Executive Summary

Predictive modeling is a powerful tool that leverages statistical techniques to forecast outcomes. It is widely used across various industries, from healthcare to finance, to make informed decisions based on historical data. This tutorial aims to introduce a framework for developing predictive models in the context of actuarial experience studies. To ground this framework within the context of real actuarial problems, we will also specifically look to understand and model the differences in mortality by product (whole life, term, etc.) With the modeling approach we can see that

The relative spread of preferred mortality differs by product.
- For 2-class preferred systems, the residual standard mortality is much higher than preferred for term than for other products.
- The spread for UL/VL/ULSG/VLSG for 4-class preferred systems is much wider than for other products.
There are divergences in the spread of face amount factors for xl, Perm, and Term, with xL narrowing relative to Term.
The issue age slope appears to be steeper for Term than Perm and xL under age 65. However, differences emerge above issue age 65, with the slope for Perm steepening relative to xL.
There are differences among the products in durations 1 and 2.
Since issues years 1990-1999, there has been a small but steady increase in relative mortality for xL vs Term, with xL now approaching Term.

Data

For what follows, we used a filtered and summarized subset of the Society of Actuaries’ Individual Life Experience Committee mortality data. Columns included in the extract were

Number of Preferred Classes
Preferred Class
Smoker Status
Face Amount Band
Observation_Year
Duration
Issue Age
Insurance Plan
Anticipated Level Term Period
Issue Year
Sex
Death Count
Death Claim Amount
Tabular Expected Mortality by Count - 2015VBT
Tabular Expected Mortality by Amount - 2015VBT

The data were filtered as

Issue ages 18 and greater
Durations 25 and less
Experience years 2013-2017

and then grouped or combined as

Underwriting: concatenation of smoker status, number of preferred classes, and preferred class, in that order
Duration: 1, 2, 3, 4-5, 6-15, 16-25
Issue Age: 18-24, 25-34, 35-44, 45-54, 55-64, 65-74, 75-84, 85-99
Issue Years: 1900-1989, 1990-1999, 2000-2009, 2010+
Insurance Plan: UL, ULSG, VL, VLSG collapsed into category “xL”
Face Amount Band: face amounts under 50,000 grouped into a single category, face amounts 1 million and greater grouped into a single category

The intent of this heavy grouping and summarization was to enable running this document with modest computing resources. The source data can be replaced with a similarly constructed dataset with finer grouped variables.

The code to generate these files can be found in the datafiles subfolder. It relies on an unpublished version of the ILEC dataset which has been restructured using the Arrow framework into a collection of Parquet files. A knowledgeable reader should be able to adapt the code to whatever environment in which they keep their own copy of the ILEC dataset.

Machine Learning in Mortality Studies

Experience studies are a primary tool that actuaries use to quantify and understand historical experience. It is a natural next step to apply statistical techniques to experience studies to discover new and relevant insights. There are multiple advantages to this approach:

Allows the actuary to avoid cumbersome and potentially misleading univariate analysis
Allows the actuary to appropriately consider credibility and unlock all the credibility inherent in the data
Makes it easier to discover and appropriately adjust for variable interactions
Enables the actuary the ability to statistically control for the different sources of variation in any given cell of a mortality study.

Problem Statement

One question of interest to actuaries is why different products exhibit different mortality outcomes. Even though they can be difficult to separately identify and quantify, it is known that underwriting, target market, policyholder behavior, and socioeconomic factors, among others, have direct bearing on mortality outcomes. With a statistical or machine learning model we have a possible solution to account for the impact of these variables. For this project, the key question we are trying to answer is how mortality varies by product in the Individual Life Experience Committee dataset. In the simplified dataset that is used herein, the product categories are Term, Perm, UL/VL, and Other. To understand the differences in mortality by product, we will construct machine learning models to predict the mortality outcomes and analyze the results for relevant insights.

Methodology

The framework will guide the process of code setup, model creation, preprocessing, and validation. It will also address common challenges often encountered such as: incorporating nonlinear relationships, determining interactions, dealing with underfitting and overfitting (bias-variance trade-off), and model interpretability. The goal of this project is to provide useful techniques, code, and ideas, to actuaries to guide future analysis of mortality studies.

There are several common key steps in any modeling process: data preprocessing, data exploration, model selection, model validation, and model interpretation. Much more can be written on these topics than we have the space to explore, and we aim to address the key considerations as they pertain to experience studies.

Modeling Approaches

When applying statistics and machine learning to experience studies, there are multiple different modeling approaches one might take. We will focus our attention on the most common approaches used: generalized linear models (GLMs), generalized linear models with penalization (also known as elastic net GLMs), and gradient boosting machines (GBMs or GBDTs). Many other approaches or variations on these approaches are also reasonable.

Generalized Linear Models (GLM)

Generalized linear models have the most history of the methods that we will examine and in some sense are the simplest. One of the benefits of GLMs is that they allow statistical hypothesis testing. For instance, individual model coefficients can be statistically tested, and various statistical tests can be performed to validate results and compare models. The results of GLMs are also relatively simple to interpret. However, GLMs have a few disadvantages: due to their relative simplicity, they have lower predictive power than other methods. To get the best performance out of a GLM, additional effort is needed to capture nonlinear relationships and interactions. Ultimately, this can make them more time-intensive than other methodologies.

GLMs can be extended into regularized GLMs, such as LASSO or Ridge, which modifies the objective used to fit the model. This regularization term offers several advantages, disadvantages, and changes to the modeling process. First, the addition of penalization makes confidence intervals and hypothesis testing infeasible. Instead of using hypothesis testing on coefficients and likelihood ratio tests to evaluate relative fitness of models, we apply a machine learning paradigm by optimizing our model using cross-validation. Fortunately, a regularized GLM still maintains the nice interpretability of a linear model, and it can increase the overall predictive accuracy of the model. Additionally, by using a LASSO penalty, it can perform automatic feature selection.

Gradient-Boosted Decision Trees (GBDT)

Gradient boosted decision trees are an ensemble of decision trees generated in a stage-wise fashion. Each decision tree is recursively trained on the residuals of the previous tree. The first tree is a decision tree on the outcome, the second the residuals on that, and so on. In this way, the model is continually refocusing on where its predictions are weakest. Popular frameworks for gradient boosted decision trees include LightGBM, CatBoost, and XGBoost. This model is one of the most effective methods for classification and regression for tabular data.

Gradient boosting machines (applied here with LightGBM) have become the go-to approach in many tabular machine learning tasks due to their very high accuracy, ease of use, and ability to seamlessly discover important interactions. However, they can also be the most complex to interpret. To aid in interpretation, we will discuss the use of SHAP values, which is a popular method of interpretation.

Model Explanation

Ordered Lorenz Plot and Gini

An ordered Lorenz curve and the associated Gini coefficient measure the ability of a model to stratify risk. An ordered Lorenz curve is created using the model prediction as an index. Using this index, we graph the cumulative percentage of claims vs the cumulative percentage of exposure. The more bowed this line, the better the model is able to predict the outcome. The Gini Index measures the difference between this line and perfect equality. The more your model is able to predict risk, the more unequal the distribution of claims is between the model prediction, and thus the larger the Gini coefficient.

Lift Plot

There are several different varieties of lift plots used in connection with machine learning. These plots are used to help visually understand the risk stratification and accuracy of a model. As presented here, lift plots sort the model predictions into deciles based upon the predicted value. For each decile, the model’s average prediction for that cell is graphed vs the value seen empirically in the data. The more these two values are in agreement, the better the model is performing.

SHAP

SHAP values are a method of model interpretation in machine learning and originally come from Shapley values in economics. SHAP values measure the impact each feature has on the prediction for a particular instance. This numeric score indicates how much each feature contributed to the prediction in terms of sign and magnitude.

Feature Importance

Feature importance is a global measure of how much a variable contributes to the predictions of a target variable within a model. This can be helpful in interpreting a model to understand the key drivers in aggregate. However, unlike SHAP values, feature importance does not help you interpret individual predictions. Feature importance is usually presented in terms of percent contribution. When done so, a feature importance of 20% for a feature would implies that 20% of the overall reduction in prediction error is attributable to that particular feature. There are multiple ways of measuring feature importance. One of the simplest and most intuitive is permutation feature importance. Using this method, you scramble a particular feature so that it is no longer useful and measure the percent difference in model performance before and after this change. The change in error would be the importance.

The reader should be cautioned that a low relative importance does not imply lack of significance or of predictive value. For example, gender is a well-known predictor of mortality. The variation explainable by other factors of the data can greatly exceed the variation arising from gender, and interactions with other variables like age can further rob gender of importance attributed to it. The effect then is to push gender down the feature importance list.

Goodness-of-Fit

No matter how well a model may behave on measures of feature importance, lift, Lorenz and Gini indices, mean square error, deviance, and so on, it is nonetheless important to check goodness-of-fit. Goodness-of-fit checks allow us to see how well a model reproduces the phenomena of interest. For our purposes, this is the same as checking ratios of actual claims to model predicted claims. In each model section, there are univariate and bivariate goodness-of-fit tables. Ideally, we should see 100% for all entries. For the GLM model and the univariate goodness-of-fit checks, we will see this throughout the tables of goodness-of-fit, as a non-penalized GLM will reproduce the margins for any included categorical variable or interaction of categorical variables. For space reasons, we omit a test for ratios significantly different from 100%. However, qualitatively, ratios far from 100%, perhaps +/- 5% or +/- 10%, should be deemed as evidence of poor fit for that cell.

Framework Preparation

Before getting to the core data analysis task, we need to first prepare the R environment by configuring display and model options, loading necessary libraries. Then, we load the data and prep for running data analysis and modeling. This section also reads in the dataset and splits the data into training and testing sets based on the observation year. Additionally, most parameters are set here. If prototyping is enabled, it creates a smaller subset of the training data for quicker processing.

Here, we set display options.

#-----------------------------------------#
##### Display Options #####
#-----------------------------------------#
## turn off scientific notation
options(scipen = 999)

## change how many digits to display
options(digits=4)

## Suppress warnings
options(warn = -1)

## Determine which output we are generating
## This will be html, docx, or pdf
if(interactive()) {
  output_format <- "html"
} else {
  output_format <- knitr::opts_knit$get("rmarkdown.pandoc.to")  
}

Here we set model options for GLMNET and LightGBM.

#-----------------------------------------#
##### Model Options #####
#-----------------------------------------#

## When TRUE, only a fraction of the data is used, drastically reducing runtimes
prototype      <- FALSE
prototype_size <- 50000  ## number of records in data fraction
nTrainSeed     <- 42     ## seed to use when splitting the data for reproducibility

## GLMNet parameters
nGLMNetCores                <- 10
nInteractionDepth           <- 1
fGLMNetAlpha                <- 0.5
nUseTopLightGBMInteractions <- "ALL" # Integer or "ALL"
bUseSparse                  <- TRUE
nELSeed                     <- 13579

## LightGBM parameters 
flgbm_vis_subset     <- 0.1
bFullInteractions    <- FALSE  # Very slow in default configuration
nPlotTopFeatures     <- 3
nPlotTopInteractions <- 3
nGBMSeed             <- 1337

## Flags for running specific models
runGLM      <- TRUE
runLightGBM <- TRUE
runGLMInt   <- TRUE

Here we load all required libraries.

#-----------------------------------------#
##### Required libraries #####
#-----------------------------------------#
## Less verbose tidyverse
options (tidyverse.quiet = TRUE)

## We use bUseGroundhog in the R codespaces to control versioning,
## If you have your own setup, set to FALSE.
## Things may nonetheless break, so use at your own risk.
bUseGroundhog <- FALSE  

if(bUseGroundhog) {
  pkgDate <- "2024-05-09"
  
  suppressPackageStartupMessages( {
      library(groundhog)
      
      groundhog.library(pre,pkgDate)
      groundhog.library(lightgbm,pkgDate)
      groundhog.library(data.table,pkgDate)  
      groundhog.library(lmtest,pkgDate)
      groundhog.library(glmnet,pkgDate)
      groundhog.library(dplyr,pkgDate)
      groundhog.library(EIX,pkgDate)
      groundhog.library(ggplot2,pkgDate)
      groundhog.library(tidyr,pkgDate)
      groundhog.library(doParallel,pkgDate)
      groundhog.library(tidyverse,pkgDate)
      groundhog.library(magrittr,pkgDate)
      groundhog.library(dtplyr,pkgDate)
      groundhog.library(flextable,pkgDate)
      groundhog.library(ftExtra,pkgDate)
      groundhog.library(arrow,pkgDate)
      groundhog.library(here,pkgDate)
      groundhog.library(shapviz,pkgDate)
      groundhog.library(patchwork,pkgDate)
      groundhog.library(Matrix,pkgDate)
      groundhog.library(MatrixModels,pkgDate)
      groundhog.library(openxlsx,pkgDate)
      groundhog.library(flexlsx,pkgDate)
    }
  )

} else {
  suppressPackageStartupMessages({
    library(pre)
    library(lightgbm)
    library(data.table)  
    library(lmtest)
    library(glmnet)
    library(dplyr)
    library(EIX)
    library(ggplot2)
    library(tidyr)
    library(doParallel)
    library(tidyverse)
    library(magrittr)
    library(dtplyr)
    library(flextable)
    library(ftExtra)
    library(arrow)
    library(here)
    library(shapviz)
    library(patchwork)
    library(Matrix)
    library(MatrixModels)
    library(openxlsx2)
    library(flexlsx)
  })
}

#-----------------------------------------#
##### Set Folder Locations #####
#-----------------------------------------#

source("R/functions.R")
source("R/glmnet_support.R")

## set library location
local_libraries <- FALSE
if(local_libraries)
{
  library.dir <- 'D:\\Data\\Niemerg\\Life Predictive Mortality POG\\01 library'
  .libPaths(new = library.dir)
}

bDebug <- FALSE

## Save the expensive working objects and reload if they exist. If this is true  
## and they do not exist, the computations will rerun.
bUseCache <- TRUE
bInvalidateCaches <- FALSE

#-----------------------------------------#
##### Data Options #####
#-----------------------------------------#

src_file <- 'http://finriskanalytics-ilecdata.s3-website-us-east-1.amazonaws.com/ilec13_17_framework_light.parquet'

cacheFileRoot <- file.path(
  getwd(),
  "objectcache",
  tools::file_path_sans_ext(
    tail(
      unlist(strsplit(src_file,"/")),
      1
    )
  )
)

exportsRoot <- file.path(
  getwd(),
  "render_exports",
  tools::file_path_sans_ext(
    tail(
      unlist(strsplit(src_file,"/")),
      1
    )
  )
)

#-----------------------------------------#
##### Modeling Parameters #####
#-----------------------------------------#
resp_var    <- "amount_actual"
resp_offset <- "amount_2015vbt"

pred_cols <- c("uw",
               "face_amount_band",
               "dur_band1",
               "ia_band1",
               "gender",
               "insurance_plan",
               "ltp",
               "iy_band1")

factor_cols <- c("uw",
               "face_amount_band",
               "dur_band1",
               "ia_band1",
               "gender",
               "insurance_plan",
               "ltp",
               "iy_band1")

Here we read the data, convert specified columns to categorical factors to ensure proper data handling and adjust the labels for the ‘face_amount_band’ factor to avoid issues in model outputs. The dataset is split into training and testing sets based on the observation year, with the year 2017 data used for validation. If prototyping is enabled, the code further subsets the training data to a smaller size for faster processing, ensuring reproducibility by setting a seed before shuffling and selecting the subset.

#-----------------------------------------#
##### Load dataset #####
#-----------------------------------------#

## Determine the file extension of the source file
file_type <- tools::file_ext(src_file)

## Read the dataset based on file extension
if (file_type == "csv") {
  ## Load a CSV file
  ds <- fread(src_file)
} else if (file_type == "parquet") {
  ## Load a Parquet file and convert it to a data table
  if(src_file %like% "amazonaws.com") {
    s3b <- s3_bucket(strsplit(urltools::domain(src_file),".",fixed=T)[[1]][1])
    ds <- read_parquet(s3b$path(urltools::path(src_file)))
  } else {
    ds <- arrow::read_parquet(src_file) %>% as.data.table()
  }
}

#-----------------------------------------#
##### Convert columns to factors #####
#-----------------------------------------#

## This step ensures that categorical data is appropriately treated as such.
ds[, (factor_cols) := lapply(.SD, factor), .SDcols = factor_cols]

## Adjust the labels for the 'face_amount_band' factor
## Replace colons in factor names with hyphens to avoid issues in model outputs
ds[, face_amount_band := fct_relabel(
  face_amount_band,
  function(x) sub(":", " -", x, fixed = TRUE)
)]

#-----------------------------------------#
##### Set train / test #####
#-----------------------------------------#

## Split the dataset into training and testing sets based on the observation year
## The year 2017 is used as the validation set
train <- ds[observation_year != 2017]
test  <- ds[observation_year == 2017]

## Subset the data for prototyping purposes
## This code block is executed if a prototype subset is requested
if (prototype) {
  ## Set the seed for reproducibility
  set.seed(nTrainSeed)
  
  ## Shuffle and select a subset of the training data
  train <- train[sample(1:nrow(train)), ]
  train <- train[1:prototype_size]
}

Models

GLM

Below is an analysis using a main-effects GLM to better understand the data. We integrate two modeling approaches:

Standard GLM analysis with model calibration (but no model building), including presentation of coefficients and residuals analysis, and
An approach to explore the interactions of the main effects model along selected dimensions of the data, checking the average main effects for those subsets, weighted according to the offset used in the analysis.

Model Summary

## Construct the model formula from predictor columns
modelFormula <- paste(pred_cols, collapse = " + ")

## Include offset in the model formula if it exists
if (exists("resp_offset")) {
  modelFormula <- paste(
    modelFormula,
    paste0("offset(log(", resp_offset, "))"),
    sep = " + "
  )
}

## Complete the model formula with the response variable
modelFormula <- paste(resp_var, modelFormula, sep = " ~ ")
modelFormula <- as.formula(modelFormula)

## Check if the cached model exists and is valid
if (bUseCache & file.exists(paste0(cacheFileRoot, "_glm_model.rds")) & !bInvalidateCaches) {
  ## Load the cached model if it exists
  modelGLM <- readRDS(paste0(cacheFileRoot, "_glm_model.rds"))
} else {
  ## Fit the GLM model using the specified formula and data
  modelGLM <- glm(
    formula = modelFormula, 
    family = quasipoisson, 
    data = train,
    x = FALSE,
    y = FALSE,
    model = FALSE
  )
  ## Remove the data from the model object to reduce size
  modelGLM$data <- c()
  
  ## Save the fitted model to cache if caching is enabled
  if (bUseCache) {
    saveRDS(modelGLM, paste0(cacheFileRoot, "_glm_model.rds"))
  }
}

## Append predictions to the dataset
ds[, predictions_glm := predict(modelGLM, newdata = .SD, type = "response")]
train[, predictions_glm := predict(modelGLM, newdata = .SD, type = "response")]
test[, predictions_glm := predict(modelGLM, newdata = .SD, type = "response")]

Model, Table of Coefficients, ANOVA

Model Summary

Below is the table of coefficients for the fitted GLM. Each entry is a coefficient in the table for the level of the indicated variable. The estimate and standard errors are on the scale of the linear predictor. For a Poisson model with log link, this means they are on the log scale.

## Show summary of the model
modelGLM %>% 
  as_flextable() %>%
  set_table_properties(opts_html=list(
    scroll=list(
      add_css="max-height: 500px;"
      )
    )
    )

	Estimate	Standard Error	z value	Pr(>\|z\|)
(Intercept)	0.978	0.091	10.743	0.0000	***
uwN/2/1	-0.152	0.011	-13.931	0.0000	***
uwN/2/2	0.248	0.010	23.945	0.0000	***
uwN/3/1	-0.324	0.014	-23.614	0.0000	***
uwN/3/2	-0.180	0.012	-14.802	0.0000	***
uwN/3/3	0.152	0.011	14.077	0.0000	***
uwN/4/1	-0.330	0.014	-23.555	0.0000	***
uwN/4/2	-0.166	0.015	-11.060	0.0000	***
uwN/4/3	0.011	0.017	0.630	0.5287
uwN/4/4	0.208	0.016	12.618	0.0000	***
uwS/1/1	0.068	0.014	4.778	0.0000	***
uwS/2/1	-0.179	0.021	-8.637	0.0000	***
uwS/2/2	0.112	0.022	5.178	0.0000	***
uwU/1/1	0.261	0.038	6.794	0.0000	***
face_amount_band04 - 50,000 - 99,999	-0.113	0.018	-6.410	0.0000	***
face_amount_band05 - 100,000 - 249,999	-0.236	0.015	-15.413	0.0000	***
face_amount_band06 - 250,000 - 499,999	-0.299	0.016	-18.893	0.0000	***
face_amount_band07 - 500,000 - 999,999	-0.322	0.016	-20.258	0.0000	***
face_amount_band08 - 1,000,000+	-0.349	0.015	-23.004	0.0000	***
dur_band102	0.010	0.030	0.341	0.7334
dur_band103	0.000	0.029	0.006	0.9951
dur_band104-05	-0.044	0.026	-1.686	0.0917	.
dur_band106-15	-0.115	0.028	-4.084	0.0000	***
dur_band116-25	-0.096	0.031	-3.088	0.0020	**
ia_band125-34	-0.062	0.034	-1.813	0.0699	.
ia_band135-44	-0.045	0.033	-1.347	0.1779
ia_band145-54	-0.075	0.033	-2.258	0.0239	*
ia_band155-64	-0.100	0.033	-3.024	0.0025	**
ia_band165-74	-0.082	0.033	-2.440	0.0147	*
ia_band175-84	-0.173	0.034	-5.100	0.0000	***
ia_band185-99	-0.304	0.040	-7.581	0.0000	***
genderM	0.011	0.006	1.790	0.0735	.
insurance_planPerm	-0.134	0.062	-2.159	0.0308	*
insurance_planTerm	-0.275	0.069	-3.977	0.0001	***
insurance_planxL	-0.042	0.061	-0.681	0.4958
ltp10 yr	-0.090	0.034	-2.650	0.0080	**
ltp15 yr	-0.097	0.034	-2.826	0.0047	**
ltp20 yr	-0.207	0.033	-6.286	0.0000	***
ltp25 yr	-0.247	0.049	-5.077	0.0000	***
ltp30 yr	-0.161	0.036	-4.502	0.0000	***
ltpNot Level Term	-0.396	0.045	-8.750	0.0000	***
ltpUnknown	-0.207	0.036	-5.804	0.0000	***
iy_band11990-1999	-0.073	0.023	-3.209	0.0013	**
iy_band12000-2009	-0.148	0.026	-5.706	0.0000	***
iy_band12010+	-0.206	0.030	-6.857	0.0000	***
Signif. codes: 0 <= '*' < 0.001 < '' < 0.01 < '*' < 0.05

(Dispersion parameter for quasipoisson family taken to be 700118)
Null deviance: 6.172e+10 on 249979 degrees of freedom
Residual deviance: 5.662e+10 on 249935 degrees of freedom

ANOVA

The ANOVA table displays the analysis of deviance for the GLM. For each variable, we see the proportion of deviance explained by that variable and its associated degrees of freedom.

## Show the ANOVA table

## Check if the cached ANOVA results exist and are valid
if (bUseCache & file.exists(paste0(cacheFileRoot, "_glm_model_anova.rds")) & !bInvalidateCaches) {
  ## Load the cached ANOVA results if they exist
  mod.glm.anova <- readRDS(paste0(cacheFileRoot, "_glm_model_anova.rds"))
} else {
  ## Perform ANOVA on the GLM model
  mod.glm.anova <- anova(modelGLM, test = "Chisq")
  
  ## Save the ANOVA results to cache if caching is enabled
  if (bUseCache) {
    saveRDS(mod.glm.anova, paste0(cacheFileRoot, "_glm_model_anova.rds"))
  }
}

## Convert the ANOVA results to a data table, add feature names, and format the table
mod.glm.anova %>% 
  as.data.table() %>%  # Convert to data table
  add_column(rownames(mod.glm.anova), .before = 1) %>%  # Add feature names as a new column
  setnames(old = 1, new = "feature") %>%  # Rename the new column to "feature"
  flextable() %>%  # Create a flextable for formatting
  set_formatter(
    `Pr(>Chi)` = function(x) ifelse(x < 0.01, "< 0.1%", sprintf("%1.2f%%", 100 * x))  # Format p-values
  )

feature	Df	Deviance	Resid. Df	Resid. Dev	Pr(>Chi)
NULL			249,979	61,716,566,139
uw	13	3,858,650,451	249,966	57,857,915,688	< 0.1%
face_amount_band	5	701,275,400	249,961	57,156,640,288	< 0.1%
dur_band1	5	118,698,616	249,956	57,037,941,672	< 0.1%
ia_band1	7	162,999,155	249,949	56,874,942,517	< 0.1%
gender	1	3,395,713	249,948	56,871,546,804	2.76%
insurance_plan	3	57,762,926	249,945	56,813,783,878	< 0.1%
ltp	7	152,501,012	249,938	56,661,282,865	< 0.1%
iy_band1	3	41,049,822	249,935	56,620,233,043	< 0.1%

Lift

The lift plot compares the GLM against the underlying mortality table.

## generate lift plot
test[,decile.table(get(resp_var),predictions_glm/get(resp_offset),get(resp_offset))] %>%
  pivot_longer(-c(decile,exposures)) %>% 
  as.data.table() %>%
  ggplot(aes(x=decile, y=value, col=name)) +  
  geom_line() +
  scale_x_continuous(breaks=c(1:10)) +
  labs(x="Decile",y="Response")+
  ggtitle("Decile Lift Plot") +
  theme_minimal() +
  scale_y_continuous(labels=scales::comma)

Lorenz Plot

The Lorenz plot demonstrates a model’s ability to stratify predictions against a null baseline.

## lorenz plot
test[,lorenz(get(resp_var), predictions_glm / get(resp_offset), get(resp_offset))]

The table of coefficients shows a number of interesting phenomena and perhaps some surprises:

Gender is not significant. Since we are using an offset of tabular expected rates, the interpretation is that the underlying differentials in the tabular expected rates are adequate for the current data, after adjusting for other factors.
Underwriting is the most influential factor from the ANOVA perspective.
Both the most recent issue years (2010+) and the most recent durations show significant mortality factors. Durations 1 and 2 are significantly higher than durations 3+
While insurance plans other than “Other” are significantly different from 0, a quick glance at the effects plot shows that the UL/VL plans are not significantly different from each other and with Perm, while Term is borderline significantly different from UL/VL.
Face amount bands 250K and greater have factors not significant from one another.

Model Illustrations and Graphics

Effects Plots

Because the model contains every column, this is equivalent to computing the marginal actual-to-tabular ratios. However, the model also provides standard errors, which is useful for assessing the significance of the marginal ratios.

## Plot amount model terms

## Calculate the dispersion parameter for the GLM model
glm_disp <- sum(modelGLM$residuals^2 * modelGLM$weights) / modelGLM$df.residual

## Generate plots for each predictor column
lapply(pred_cols, function(s) {
  ## Summarize the data for the current predictor
  p <- ds[, .(
      predicted = sum(predictions_glm) / sum(amount_2015vbt),
      stde = sqrt(sum(predictions_glm) * glm_disp) / sum(amount_2015vbt)
    ), by = c(s)] %>%
    setnames(s, "x") %>%  # Rename the grouping column to "x"
    mutate(
      x = fct_relevel(
        x,
        sort(levels(x))  # Reorder factor levels
      )
    ) %>%
    as.data.table() %>%
    ## Create the plot
    ggplot(aes(x = x, y = predicted)) +
    geom_point() +  # Add points for the predicted values
    geom_errorbar(
      aes(
        ymin = predicted - 1.96 * stde,
        ymax = predicted + 1.96 * stde
      )  # Add error bars for the 95% confidence interval
    ) +
    geom_hline(yintercept = 1, linetype = 2) +  # Add a horizontal line at y = 1
    scale_y_continuous(
      name = "Factor",
      labels = scales::percent  # Format y-axis labels as percentages
    ) +
    scale_x_discrete(name = s) +  # Set x-axis name to the current predictor
    theme_minimal() +  # Use a minimal theme for the plot
    theme(
      axis.text.x = element_text(angle = 45)  # Rotate x-axis text for readability
    )
  
  return(p)  # Return the plot
}) %>%
  purrr::set_names(pred_cols) %>%  # Set names for each plot based on predictor columns
  iwalk(~ {
    cat('##### ', .y, '\n\n')  # Print the plot title
    print(.x)  # Print the plot
    cat('\n\n')  # Add spacing after each plot
  })

uw

face_amount_band

dur_band1

ia_band1

gender

insurance_plan

ltp

iy_band1

Goodness-of-Fit

Goodness-of-fit tables are provided. Each table provides actual-to-model ratios for single variables and for 2-way combinations of variables. A model is qualitatively deemed to perform well if goodness-of-fit ratios are close to 100% in almost all situations. The quantitative assessment using significance testing is omitted here.

Unvariate Goodness-of-Fit

## Generate summary tables for each factor column and format them
map(factor_cols, .f = \(x) {
  ## Convert column name to symbol for tidy evaluation
  x <- sym(x)
  resp_var_sym <- resp_var
  
  ## Summarize data by the current factor column
  train %>%
    group_by(!!x) %>%
    summarize(
      Outcome = sum(amount_actual),  # Sum the actual amounts
      AM = sum(amount_actual) / sum(predictions_glm)  # Calculate the Actual-to-Model ratio
    ) %>%
    ## Create a flextable for the summarized data
    flextable() %>%
    ## Format the Actual-to-Model column as percentages
    set_formatter(
      AM = function(x) {
        if (is.numeric(x))
          sprintf("%.1f%%", x * 100)
        else
          x
      }
    ) %>%
    ## Format the Outcome column as numbers
    colformat_num(j = "Outcome") %>%
    ## Set header labels for the table
    set_header_labels(
      Outcome = "Outcome",
      AM = "Actual-to-Model"
    ) %>%
    autofit() # %>%
    ## Print the flextable
    #knitr::knit_print()
}) %>%
  ## Set names for each table based on the factor columns
  purrr::set_names(factor_cols) -> 
  output_tables

if(output_format == "html") {
  output_tables %>%
    ## Print the flextable
    map(.f=knitr::knit_print) %>%
    ## Generate a tabset from the list of tables
    generate_tabset(
      tabtitle = "",
      tablevel = 5
    ) %>%
    ## Print the generated tabset
    cat()  
} else {
  export_tables_to_excel(
    output_tables,
    paste0(
        exportsRoot,
        "_glm_univariate_goodness_of_fit_tables.xlsx"
      )
  )
  
  cat("See included Excel table for additional information.\n")
}

uw

uw	Outcome	Actual-to-Model
N/1/1	19,203,624,009	100.0%
N/2/1	9,661,768,053	100.0%
N/2/2	11,546,531,146	100.0%
N/3/1	5,950,371,727	100.0%
N/3/2	8,810,968,555	100.0%
N/3/3	14,970,350,057	100.0%
N/4/1	6,842,959,121	100.0%
N/4/2	5,162,511,227	100.0%
N/4/3	3,565,224,892	100.0%
N/4/4	3,742,908,023	100.0%
S/1/1	4,365,025,076	100.0%
S/2/1	1,910,521,866	100.0%
S/2/2	1,704,933,729	100.0%
U/1/1	498,038,979	100.0%

face_amount_band

face_amount_band	Outcome	Actual-to-Model
01 - 0 - 49,999	4,193,809,072	100.0%
04 - 50,000 - 99,999	5,283,024,574	100.0%
05 - 100,000 - 249,999	16,241,007,604	100.0%
06 - 250,000 - 499,999	14,745,604,133	100.0%
07 - 500,000 - 999,999	15,180,744,107	100.0%
08 - 1,000,000+	42,291,546,970	100.0%

dur_band1

dur_band1	Outcome	Actual-to-Model
01	1,269,136,295	100.0%
02	1,852,767,081	100.0%
03	2,494,157,892	100.0%
04-05	6,321,445,332	100.0%
06-15	55,677,490,288	100.0%
16-25	30,320,739,572	100.0%

ia_band1

ia_band1	Outcome	Actual-to-Model
18-24	667,931,081	100.0%
25-34	6,267,527,815	100.0%
35-44	16,990,453,137	100.0%
45-54	20,629,510,315	100.0%
55-64	19,447,152,090	100.0%
65-74	16,608,837,516	100.0%
75-84	15,884,011,950	100.0%
85-99	1,440,312,556	100.0%

gender

gender	Outcome	Actual-to-Model
F	33,257,191,935	100.0%
M	64,678,544,525	100.0%

insurance_plan

insurance_plan	Outcome	Actual-to-Model
Other	187,352,300	100.0%
Perm	12,161,923,261	100.0%
Term	39,023,616,053	100.0%
xL	46,562,844,846	100.0%

ltp

ltp	Outcome	Actual-to-Model
5 yr	686,343,514	100.0%
10 yr	7,271,664,723	100.0%
15 yr	6,017,756,560	100.0%
20 yr	17,767,048,607	100.0%
25 yr	525,222,922	100.0%
30 yr	3,092,731,785	100.0%
Not Level Term	59,614,315,535	100.0%
Unknown	2,960,652,814	100.0%

iy_band1

iy_band1	Outcome	Actual-to-Model
1900-1989	1,464,179,514	100.0%
1990-1999	29,544,147,346	100.0%
2000-2009	55,045,293,308	100.0%
2010+	11,882,116,292	100.0%

Bivariate Goodness-of-Fit

## Generate a list of unique pairs of factor columns
pairlist <- data.table()
for (i in 1:(length(factor_cols) - 1)) {
  for (j in (i + 1):length(factor_cols)) {
    ## Initialize or append to the pairlist
    if (i == 1 & j == 2) {
      pairlist <- data.table(F1 = factor_cols[i], F2 = factor_cols[j])
    } else {
      pairlist <- rbind(pairlist, data.table(F1 = factor_cols[i], F2 = factor_cols[j]))
    }
  }
}

## Generate and format summary tables for each pair of factor columns
map2(.x = pairlist$F1, .y = pairlist$F2, .f = \(x, y) {
  xs <- sym(x)
  ys <- sym(y)
  
  ## Choose grouping order based on the number of levels in each factor
  if (length(train[, levels(get(x))]) >= length(train[, levels(get(y))])) {
    fttmp <- train %>%
      group_by(!!xs, !!ys) %>%
      summarize(
        Outcome = sum(amount_actual),
        Ratio = sprintf("%.1f%%", 100 * sum(amount_actual) / sum(predictions_glm))
      ) %>%
      pivot_wider(
        names_from = !!ys,
        values_from = c(Outcome, Ratio),
        names_glue = paste0(y, ": {", y, "}.{.value}"),
        names_vary = "slowest"
      )
  } else {
    fttmp <- train %>%
      group_by(!!ys, !!xs) %>%
      summarize(
        Outcome = sum(amount_actual),
        Ratio = sprintf("%.1f%%", 100 * sum(amount_actual) / sum(predictions_glm))
      ) %>%
      pivot_wider(
        names_from = !!xs,
        values_from = c(Outcome, Ratio),
        names_glue = paste0(x, ": {", x, "}.{.value}"),
        names_vary = "slowest"
      )
  }
  
  ## Adjust column keys for the flextable
  fttmp.colkeys <- names(fttmp)[1]
  for (i in 1:((length(names(fttmp)) - 1) / 2)) {
    fttmp.colkeys <- c(fttmp.colkeys, paste0("blank", i), names(fttmp)[(2 * i):(2 * i + 1)])
  }
  
  ## Create and print the flextable
  fttmp %>%
    flextable(col_keys = fttmp.colkeys) %>%
    ftExtra::span_header(sep = "\\.") %>%
    align(align = 'center', part = "all") %>%
    empty_blanks() %>%
    autofit() #%>%
    #knitr::knit_print()
}) %>%
  ## Set names for each element in the list based on the factor column pairs
  purrr::set_names(pairlist[, paste0(F1, " x ", F2)]) ->
  output_tables

if(output_format == "html") {
  output_tables %>%
    map(.f=knitr::knit_print) %>%
    ## Generate a tabset from the list of formatted tables
    generate_tabset(tabtitle = "", tablevel = 5) %>%
    ## Print the generated tabset
    cat()
} else {
  export_tables_to_excel(
    output_tables,
    paste0(
        exportsRoot,
        "_glm_bivariate_goodness_of_fit_tables.xlsx"
      )
  )
  
  cat("See included Excel table for additional information.\n")
}

uw x face_amount_band

uw	face_amount_band: 01 - 0 - 49,999		face_amount_band: 04 - 50,000 - 99,999		face_amount_band: 05 - 100,000 - 249,999		face_amount_band: 06 - 250,000 - 499,999		face_amount_band: 07 - 500,000 - 999,999		face_amount_band: 08 - 1,000,000+
uw	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
N/1/1	2,331,609,989	99.5%	2,665,573,911	99.3%	4,377,227,666	98.0%	2,484,222,240	97.0%	2,280,087,941	97.1%	5,064,902,262	105.6%
N/2/1	67,510,912	104.4%	227,491,364	103.6%	1,877,102,924	100.7%	1,581,998,866	100.7%	1,613,477,143	101.9%	4,294,186,844	98.5%
N/2/2	478,620,100	97.9%	716,716,216	101.7%	2,460,749,546	101.5%	1,807,687,840	99.4%	1,536,284,872	98.7%	4,546,472,572	99.9%
N/3/1	4,589,248	123.5%	36,548,841	109.3%	636,379,622	103.7%	1,099,392,788	101.8%	1,269,934,939	99.7%	2,903,526,289	98.6%
N/3/2	15,879,147	108.5%	84,477,290	113.1%	824,647,078	103.7%	1,194,151,445	104.1%	1,294,166,133	101.7%	5,397,647,462	98.0%
N/3/3	89,078,013	102.7%	266,651,094	95.5%	1,685,052,988	100.4%	1,897,715,243	100.0%	1,940,524,074	101.9%	9,091,328,645	99.6%
N/4/1	2,890,726	85.7%	23,168,994	98.8%	533,862,585	98.0%	1,152,655,538	101.6%	1,611,064,623	98.0%	3,519,316,655	100.8%
N/4/2	5,630,152	92.5%	43,592,461	100.9%	580,655,929	100.2%	959,152,506	106.4%	1,142,130,198	101.0%	2,431,349,981	97.2%
N/4/3	2,758,598	97.9%	19,493,024	95.6%	410,982,168	104.1%	652,807,204	97.6%	822,151,439	106.1%	1,657,032,459	97.3%
N/4/4	16,764,842	89.3%	69,558,070	97.5%	501,578,433	96.3%	668,382,546	96.0%	698,077,315	99.6%	1,788,546,817	103.1%
S/1/1	846,343,572	100.2%	864,666,577	100.8%	1,161,486,983	98.4%	510,128,489	98.7%	361,498,469	88.0%	620,900,986	112.0%
S/2/1	21,486,405	106.5%	85,321,233	108.0%	614,066,701	103.4%	400,530,746	99.9%	315,167,594	101.5%	473,949,187	93.6%
S/2/2	74,338,988	105.5%	135,934,467	97.3%	492,786,516	99.0%	294,838,420	97.5%	267,040,410	113.8%	439,994,928	95.6%
U/1/1	236,308,380	105.1%	43,831,032	84.9%	84,428,465	95.8%	41,940,262	87.0%	29,138,957	82.8%	62,391,883	124.6%

uw x dur_band1

uw	dur_band1: 01		dur_band1: 02		dur_band1: 03		dur_band1: 04-05		dur_band1: 06-15		dur_band1: 16-25
uw	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
N/1/1	107,404,206	95.6%	127,611,866	107.1%	186,488,501	112.4%	495,415,170	119.6%	4,962,783,774	101.2%	13,323,920,492	98.8%
N/2/1	38,939,578	65.2%	117,924,754	143.0%	103,706,358	94.4%	187,685,672	70.1%	4,423,311,154	96.1%	4,790,200,537	105.6%
N/2/2	68,820,764	105.6%	89,429,859	102.0%	153,874,204	132.7%	321,513,019	104.9%	6,248,105,591	98.1%	4,664,787,709	101.4%
N/3/1	91,917,216	87.5%	162,393,895	101.4%	239,046,418	113.4%	463,263,819	94.1%	4,110,289,163	102.1%	883,461,216	92.5%
N/3/2	73,068,462	88.3%	142,447,178	110.4%	172,423,532	97.4%	471,435,974	96.3%	7,112,246,310	100.0%	839,347,099	101.9%
N/3/3	209,144,931	114.7%	274,764,174	100.2%	374,385,325	102.6%	1,050,466,173	109.0%	12,379,149,521	99.7%	682,439,933	88.3%
N/4/1	174,316,019	96.2%	228,792,240	79.4%	389,498,840	97.6%	1,034,633,390	104.2%	4,624,074,611	100.7%	391,644,021	100.5%
N/4/2	115,650,082	104.0%	173,110,541	98.6%	196,392,926	78.7%	696,533,900	100.7%	3,717,590,891	101.6%	263,232,887	95.8%
N/4/3	97,264,439	82.8%	192,504,643	110.5%	206,119,588	89.9%	501,287,156	91.0%	2,450,944,955	103.1%	117,104,111	101.4%
N/4/4	153,464,379	111.5%	195,703,696	94.8%	283,447,385	103.2%	660,958,667	97.3%	2,334,500,075	101.7%	114,833,821	76.4%
S/1/1	30,145,775	147.3%	28,847,772	106.8%	40,393,654	116.2%	91,257,938	102.5%	1,121,084,058	104.1%	3,053,295,879	98.0%
S/2/1	42,753,182	111.3%	58,621,979	100.4%	74,856,996	95.9%	177,694,957	94.5%	1,089,203,131	94.4%	467,391,621	118.9%
S/2/2	55,705,006	124.3%	55,852,780	86.4%	68,480,349	87.3%	160,693,564	87.0%	985,575,007	99.4%	378,627,023	111.0%
U/1/1	10,542,256	100.1%	4,761,704	82.7%	5,043,816	87.2%	8,605,933	76.0%	118,632,047	114.3%	350,453,223	97.1%

uw x ia_band1

uw	ia_band1: 18-24		ia_band1: 25-34		ia_band1: 35-44		ia_band1: 45-54		ia_band1: 55-64		ia_band1: 65-74		ia_band1: 75-84		ia_band1: 85-99
uw	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
N/1/1	195,107,686	96.3%	1,468,766,511	91.2%	3,219,766,527	91.3%	3,573,783,086	93.9%	4,457,649,299	103.5%	4,148,815,415	106.2%	1,980,748,076	115.6%	158,987,409	120.4%
N/2/1	87,770,402	105.3%	804,579,965	97.8%	1,676,447,802	93.7%	1,660,839,957	94.5%	1,585,401,151	96.2%	2,098,151,266	103.6%	1,649,244,265	111.5%	99,333,245	172.7%
N/2/2	82,359,855	103.6%	734,969,884	107.3%	1,537,827,224	100.9%	1,935,817,830	104.4%	1,907,603,917	99.1%	2,320,063,956	98.1%	2,742,077,451	97.5%	285,811,029	94.1%
N/3/1	42,971,227	94.3%	542,310,521	103.9%	1,509,846,536	99.2%	1,517,047,115	99.4%	1,083,241,294	99.5%	682,592,797	96.6%	557,962,237	106.1%	14,400,000	104.5%
N/3/2	26,380,775	131.2%	339,627,773	112.9%	1,177,249,350	110.7%	1,552,812,212	104.2%	1,501,263,082	98.8%	1,589,866,999	94.0%	2,421,887,776	94.6%	201,880,588	122.9%
N/3/3	29,150,024	93.3%	391,634,706	111.4%	1,364,327,286	116.9%	1,932,181,705	105.6%	2,306,037,454	102.7%	2,995,484,451	100.0%	5,412,062,505	94.2%	539,471,926	89.2%
N/4/1	26,287,124	77.6%	602,385,596	102.4%	2,097,046,161	105.1%	2,253,038,301	102.5%	1,386,295,039	92.5%	431,665,885	96.8%	43,983,146	55.2%	2,257,869	129.5%
N/4/2	9,905,787	101.6%	233,732,519	96.1%	1,112,629,077	104.3%	1,595,224,194	97.4%	1,413,399,308	100.2%	448,165,240	84.4%	313,744,476	145.1%	35,710,626	76.9%
N/4/3	7,030,001	83.1%	209,711,793	120.4%	705,969,366	99.8%	1,147,207,102	104.5%	986,258,205	98.7%	356,988,649	88.4%	140,290,698	90.0%	11,769,078	65.1%
N/4/4	9,949,832	98.6%	167,099,067	107.9%	638,256,939	107.0%	1,003,674,951	104.2%	958,007,317	95.6%	551,614,283	95.5%	345,232,117	97.4%	69,073,517	82.5%
S/1/1	68,766,581	104.2%	451,737,170	93.4%	1,077,472,651	95.2%	1,226,674,054	96.4%	936,351,195	110.9%	461,642,247	100.8%	139,279,941	131.2%	3,101,237	80.4%
S/2/1	26,876,275	144.2%	179,096,556	93.2%	474,899,912	95.3%	608,218,812	101.6%	402,893,657	101.2%	174,002,125	107.8%	42,414,641	99.9%	2,119,888	278.7%
S/2/2	17,471,313	129.9%	117,414,041	101.0%	333,323,589	97.3%	526,461,474	103.9%	400,538,301	90.6%	233,737,976	109.2%	63,750,567	101.0%	12,236,468	183.8%
U/1/1	37,904,199	83.5%	24,461,713	107.5%	65,390,717	112.1%	96,529,522	103.3%	122,212,871	105.3%	116,046,227	91.7%	31,334,054	96.9%	4,159,676	132.3%

uw x gender

uw	gender: F		gender: M
uw	Outcome	Ratio	Outcome	Ratio
N/1/1	6,871,313,340	100.8%	12,332,310,669	99.6%
N/2/1	3,739,559,716	101.2%	5,922,208,337	99.3%
N/2/2	4,668,956,646	100.5%	6,877,574,500	99.6%
N/3/1	1,982,219,430	100.8%	3,968,152,297	99.6%
N/3/2	3,036,089,596	97.7%	5,774,878,959	101.3%
N/3/3	5,764,067,198	101.9%	9,206,282,859	98.8%
N/4/1	1,822,625,798	98.2%	5,020,333,323	100.7%
N/4/2	997,342,369	90.9%	4,165,168,858	102.5%
N/4/3	561,153,990	89.4%	3,004,070,902	102.3%
N/4/4	894,420,215	96.9%	2,848,487,808	101.0%
S/1/1	1,618,467,456	102.5%	2,746,557,620	98.6%
S/2/1	534,567,286	96.4%	1,375,954,580	101.5%
S/2/2	553,360,607	107.2%	1,151,573,122	96.9%
U/1/1	213,048,288	98.0%	284,990,691	101.5%

uw x insurance_plan

uw	insurance_plan: Other		insurance_plan: Perm		insurance_plan: Term		insurance_plan: xL
uw	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
N/1/1	41,338,662	134.5%	6,000,364,392	97.6%	3,545,626,085	89.4%	9,616,294,870	106.2%
N/2/1	10,081,665	86.4%	1,335,375,364	96.1%	2,232,053,273	96.2%	6,084,257,751	102.4%
N/2/2	15,784,451	80.0%	1,659,464,730	101.6%	3,001,480,499	109.6%	6,869,801,466	96.0%
N/3/1	40,123,942	295.6%	317,915,492	140.1%	3,500,287,255	96.0%	2,092,045,038	101.5%
N/3/2	20,508,375	105.8%	282,009,643	131.8%	3,266,885,153	105.3%	5,241,565,384	95.7%
N/3/3	32,085,711	110.3%	496,907,377	134.1%	4,052,652,325	100.8%	10,388,704,644	98.5%
N/4/1	2,300,000	31.2%	9,330,053	136.5%	6,251,798,169	101.0%	579,530,899	90.5%
N/4/2	7,223,579	59.0%	7,940,683	100.1%	4,382,666,145	100.3%	764,680,820	98.9%
N/4/3	103,684	1.9%	2,495,706	66.5%	3,036,110,841	101.6%	526,514,661	93.0%
N/4/4	3,739,130	20.3%	12,550,388	101.1%	2,742,467,039	101.1%	984,151,466	98.5%
S/1/1	3,801,542	137.6%	1,397,404,613	90.5%	1,062,735,519	108.1%	1,901,083,402	103.6%
S/2/1	6,544,922	105.9%	163,842,775	103.1%	1,131,185,266	99.2%	608,948,903	100.7%
S/2/2	3,059,454	31.6%	189,503,330	110.6%	778,511,625	97.7%	733,859,320	100.9%
U/1/1	657,183	67.3%	286,818,715	105.3%	39,156,859	82.2%	171,406,222	96.7%

uw x ltp

uw	ltp: 5 yr		ltp: 10 yr		ltp: 15 yr		ltp: 20 yr		ltp: 25 yr		ltp: 30 yr		ltp: Not Level Term		ltp: Unknown
uw	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
N/1/1	167,693,145	109.1%	329,259,367	87.7%	313,430,210	105.4%	1,254,368,612	83.3%	22,798,990	117.5%	140,376,344	88.0%	15,709,773,948	102.8%	1,265,923,393	89.9%
N/2/1	109,944,549	104.3%	251,674,560	102.1%	154,929,926	135.1%	1,003,102,210	89.8%	54,969,322	93.3%	91,526,385	94.7%	7,455,632,156	101.2%	539,988,945	97.8%
N/2/2	101,292,185	97.6%	464,466,296	118.6%	319,758,321	113.0%	1,222,999,988	106.1%	275,278,999	105.5%	237,099,784	102.1%	8,573,983,079	96.9%	351,652,494	127.0%
N/3/1	41,020,000	109.2%	402,661,935	102.2%	437,170,279	100.8%	1,876,102,407	92.5%	32,892,500	81.8%	371,739,248	91.5%	2,700,444,573	106.6%	88,340,785	115.1%
N/3/2	62,023,000	152.0%	523,988,195	109.0%	476,579,573	99.7%	1,824,845,270	105.8%	12,110,000	59.7%	210,771,725	91.7%	5,612,039,277	97.2%	88,611,515	141.4%
N/3/3	59,980,899	76.7%	896,107,443	101.3%	652,611,790	96.4%	1,842,591,351	104.0%	16,475,000	92.4%	201,629,249	93.7%	11,161,374,076	99.6%	139,580,249	120.4%
N/4/1	4,410,000	189.8%	1,142,209,199	105.5%	1,124,192,978	94.8%	3,080,000,938	101.6%	36,372,000	125.8%	845,189,202	101.3%	591,162,804	90.3%	19,422,000	84.8%
N/4/2	985,000	47.5%	899,393,774	97.8%	918,062,758	97.6%	2,118,167,322	102.4%	14,432,100	97.5%	414,673,900	103.8%	779,845,082	98.3%	16,951,291	73.2%
N/4/3	300,000	40.4%	711,850,279	85.6%	663,787,225	104.9%	1,348,193,434	106.8%	8,775,000	155.8%	270,343,655	116.8%	529,114,051	91.9%	32,861,248	130.8%
N/4/4	1,900,000	70.6%	861,440,026	96.0%	549,878,289	100.2%	1,117,119,304	106.1%	8,832,928	95.4%	186,021,492	107.1%	1,000,440,984	97.2%	17,275,000	61.2%
S/1/1	77,649,707	94.9%	220,477,191	117.0%	92,140,018	95.6%	325,974,726	108.2%	37,348,584	91.1%	43,126,718	114.7%	3,324,217,468	97.7%	244,090,664	113.2%
S/2/1	35,606,411	76.5%	312,687,204	100.7%	190,844,063	101.1%	450,793,441	99.4%	2,925,000	103.9%	50,493,377	107.2%	784,528,034	101.3%	82,644,336	95.5%
S/2/2	19,448,975	76.3%	251,942,737	95.5%	122,991,130	92.6%	294,652,404	102.4%	1,050,000	59.2%	28,434,706	103.2%	932,515,030	101.8%	53,898,747	108.9%
U/1/1	4,089,643	79.6%	3,506,517	78.9%	1,380,000	19.3%	8,137,200	103.2%	962,499	27.8%	1,306,000	73.8%	459,244,973	101.9%	19,412,147	111.7%

uw x iy_band1

uw	iy_band1: 1900-1989		iy_band1: 1990-1999		iy_band1: 2000-2009		iy_band1: 2010+
uw	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
N/1/1	982,160,538	101.0%	12,534,732,161	98.5%	4,842,664,276	101.9%	844,067,034	111.3%
N/2/1	112,802,132	111.6%	4,655,974,027	105.8%	4,443,736,468	95.7%	449,255,426	86.6%
N/2/2	107,232,938	97.9%	4,512,424,274	101.4%	6,312,557,239	98.2%	614,316,695	110.2%
N/3/1	282,000	25.0%	1,044,203,844	94.0%	3,887,317,728	101.0%	1,018,568,155	102.9%
N/3/2	250,000	15.7%	1,008,204,301	108.1%	6,934,988,157	99.2%	867,526,097	98.0%
N/3/3	4,055,561	29.7%	755,727,777	89.7%	12,308,187,396	100.0%	1,902,379,323	105.0%
N/4/1	0	0.0%	394,664,512	92.8%	4,606,160,597	101.2%	1,842,134,012	98.8%
N/4/2			308,834,925	92.1%	3,702,484,884	102.2%	1,151,191,418	95.6%
N/4/3			158,854,990	106.6%	2,386,096,189	101.6%	1,020,273,713	95.6%
N/4/4	0	0.0%	146,243,472	85.1%	2,314,552,528	101.3%	1,282,112,023	99.7%
S/1/1	212,607,764	96.3%	2,836,569,692	97.3%	1,124,908,213	106.3%	190,939,407	111.3%
S/2/1	7,889,930	105.5%	479,056,419	117.8%	1,077,182,521	95.0%	346,392,996	95.7%
S/2/2	7,665,540	118.1%	388,469,845	110.5%	986,492,416	100.9%	322,305,928	87.3%
U/1/1	29,233,111	102.7%	320,187,107	96.6%	117,964,696	113.5%	30,654,065	89.3%

face_amount_band x dur_band1

face_amount_band	dur_band1: 01		dur_band1: 02		dur_band1: 03		dur_band1: 04-05		dur_band1: 06-15		dur_band1: 16-25
face_amount_band	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
01 - 0 - 49,999	37,859,834	206.5%	46,746,236	170.9%	59,349,580	153.9%	129,333,013	132.3%	856,334,359	114.3%	3,064,186,050	93.9%
04 - 50,000 - 99,999	52,511,061	154.8%	66,787,618	135.5%	90,269,149	137.0%	206,424,919	123.5%	1,469,057,939	106.5%	3,397,973,888	94.7%
05 - 100,000 - 249,999	191,101,070	116.3%	269,652,985	110.7%	371,980,209	113.2%	855,544,316	103.2%	7,074,965,104	102.1%	7,477,763,920	96.6%
06 - 250,000 - 499,999	209,469,031	101.1%	304,453,835	99.9%	416,690,820	102.2%	1,000,438,105	98.2%	8,109,440,540	102.1%	4,705,111,802	96.7%
07 - 500,000 - 999,999	215,479,316	89.6%	350,555,118	99.6%	461,397,659	96.9%	1,167,345,786	98.3%	8,791,983,362	100.6%	4,193,982,866	100.3%
08 - 1,000,000+	562,715,983	93.0%	814,571,289	93.0%	1,094,470,475	93.0%	2,962,359,193	98.1%	29,375,708,984	98.1%	7,481,721,046	112.0%

face_amount_band x ia_band1

ia_band1	face_amount_band: 01 - 0 - 49,999		face_amount_band: 04 - 50,000 - 99,999		face_amount_band: 05 - 100,000 - 249,999		face_amount_band: 06 - 250,000 - 499,999		face_amount_band: 07 - 500,000 - 999,999		face_amount_band: 08 - 1,000,000+
ia_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
18-24	75,552,784	113.3%	133,231,819	104.9%	220,299,419	96.4%	115,618,404	90.3%	61,754,428	95.5%	61,474,227	115.8%
25-34	192,799,300	135.4%	458,412,108	108.4%	1,476,938,558	99.2%	1,398,298,659	97.1%	1,378,321,244	99.1%	1,362,757,946	98.6%
35-44	421,318,794	111.8%	898,534,810	103.5%	3,004,123,357	98.4%	3,258,823,688	98.7%	3,627,163,035	96.5%	5,780,489,453	102.6%
45-54	861,129,196	105.6%	1,234,583,452	103.1%	4,027,597,384	100.6%	3,776,175,782	98.0%	3,750,396,726	96.9%	6,979,627,775	101.3%
55-64	1,371,789,823	97.4%	1,302,388,605	98.1%	4,041,501,912	100.2%	3,254,795,121	100.4%	2,876,205,825	98.5%	6,600,470,804	101.2%
65-74	1,035,580,594	89.8%	950,356,588	92.1%	2,489,528,859	99.8%	1,929,056,738	103.8%	2,107,668,820	107.8%	8,096,645,917	99.8%
75-84	221,626,122	101.6%	285,327,929	97.6%	901,837,477	103.1%	915,794,848	108.8%	1,257,117,333	112.8%	12,302,308,241	98.1%
85-99	14,012,459	119.7%	20,189,263	120.5%	79,180,638	118.3%	97,040,893	115.0%	122,116,696	115.6%	1,107,772,607	95.9%

face_amount_band x gender

face_amount_band	gender: F		gender: M
face_amount_band	Outcome	Ratio	Outcome	Ratio
01 - 0 - 49,999	2,082,160,405	95.0%	2,111,648,667	105.4%
04 - 50,000 - 99,999	2,075,208,647	94.5%	3,207,815,927	103.9%
05 - 100,000 - 249,999	5,592,847,725	95.7%	10,648,159,879	102.4%
06 - 250,000 - 499,999	4,715,664,348	99.9%	10,029,939,785	100.1%
07 - 500,000 - 999,999	4,435,784,376	102.6%	10,744,959,731	99.0%
08 - 1,000,000+	14,355,526,434	102.7%	27,936,020,536	98.7%

face_amount_band x insurance_plan

face_amount_band	insurance_plan: Other		insurance_plan: Perm		insurance_plan: Term		insurance_plan: xL
face_amount_band	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
01 - 0 - 49,999	4,987,841	93.2%	2,341,324,220	98.6%	430,107,626	122.3%	1,417,389,385	97.0%
04 - 50,000 - 99,999	5,666,783	99.9%	1,269,348,289	93.3%	1,013,673,047	124.2%	2,994,336,455	96.6%
05 - 100,000 - 249,999	34,041,514	117.9%	2,315,213,211	95.1%	6,874,173,321	102.8%	7,017,579,558	99.0%
06 - 250,000 - 499,999	28,639,038	138.2%	1,510,214,779	94.3%	8,348,184,153	99.6%	4,858,566,163	102.4%
07 - 500,000 - 999,999	18,132,026	99.2%	1,434,005,349	91.2%	8,575,363,506	97.6%	5,153,243,226	107.3%
08 - 1,000,000+	95,885,098	88.4%	3,291,817,413	116.8%	13,782,114,400	98.4%	25,121,730,059	99.0%

face_amount_band x ltp

ltp	face_amount_band: 01 - 0 - 49,999		face_amount_band: 04 - 50,000 - 99,999		face_amount_band: 05 - 100,000 - 249,999		face_amount_band: 06 - 250,000 - 499,999		face_amount_band: 07 - 500,000 - 999,999		face_amount_band: 08 - 1,000,000+
ltp	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
5 yr	40,106,609	99.0%	108,758,076	109.8%	223,852,354	96.5%	134,040,438	98.9%	88,356,037	106.2%	91,230,000	95.0%
10 yr	79,471,212	114.9%	152,307,148	128.4%	1,212,386,424	106.6%	1,312,587,882	102.0%	1,417,042,517	102.6%	3,097,869,540	94.5%
15 yr	63,154,393	123.8%	171,454,282	118.5%	1,115,915,003	101.2%	1,284,208,527	99.8%	1,254,669,304	98.2%	2,128,355,051	98.8%
20 yr	113,505,987	126.9%	278,179,336	125.6%	2,893,988,498	100.6%	3,894,785,171	99.7%	4,168,688,693	97.7%	6,417,900,922	100.1%
25 yr	9,580,391	167.3%	26,792,196	146.7%	154,206,000	105.4%	162,940,708	96.8%	101,887,627	87.9%	69,816,000	98.8%
30 yr	6,329,793	185.7%	31,267,129	163.5%	436,286,383	109.2%	721,667,111	98.6%	809,282,434	96.0%	1,087,898,935	99.3%
Not Level Term	3,802,855,700	98.1%	4,317,239,186	95.6%	9,477,900,932	98.0%	6,615,106,546	100.5%	6,758,480,601	103.0%	28,642,732,570	100.8%
Unknown	78,804,987	138.5%	197,027,221	133.2%	726,472,010	107.1%	620,267,750	95.4%	582,336,894	89.1%	755,743,952	97.7%

face_amount_band x iy_band1

face_amount_band	iy_band1: 1900-1989		iy_band1: 1990-1999		iy_band1: 2000-2009		iy_band1: 2010+
face_amount_band	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
01 - 0 - 49,999	267,683,109	88.3%	2,808,814,596	94.6%	833,122,239	113.9%	284,189,128	150.3%
04 - 50,000 - 99,999	267,216,839	93.3%	3,149,056,719	94.8%	1,444,089,389	106.9%	422,661,627	130.8%
05 - 100,000 - 249,999	404,345,538	94.5%	7,205,856,453	96.5%	6,896,526,088	102.1%	1,734,279,525	109.1%
06 - 250,000 - 499,999	158,023,865	94.6%	4,708,333,873	97.2%	7,919,827,744	101.7%	1,959,418,651	100.8%
07 - 500,000 - 999,999	136,967,257	113.5%	4,219,050,921	100.2%	8,611,625,509	100.4%	2,213,100,420	97.6%
08 - 1,000,000+	229,942,906	144.9%	7,453,034,784	110.8%	29,340,102,339	98.3%	5,268,466,941	94.6%

dur_band1 x ia_band1

ia_band1	dur_band1: 01		dur_band1: 02		dur_band1: 03		dur_band1: 04-05		dur_band1: 06-15		dur_band1: 16-25
ia_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
18-24	38,014,843	88.5%	30,257,763	81.9%	27,602,930	89.9%	48,778,652	96.0%	247,658,415	121.0%	275,618,478	91.3%
25-34	134,511,109	103.4%	163,517,330	108.7%	194,704,716	111.7%	424,147,771	112.2%	2,807,372,903	100.3%	2,543,273,986	96.5%
35-44	290,877,481	135.5%	372,345,417	117.1%	502,388,214	117.5%	1,120,149,604	104.8%	8,625,261,989	101.2%	6,079,430,432	94.4%
45-54	284,754,759	87.7%	443,414,694	93.4%	668,187,442	103.0%	1,648,875,363	101.9%	10,888,781,617	102.0%	6,695,496,440	97.2%
55-64	316,638,372	94.6%	537,383,357	107.2%	610,988,925	88.1%	1,580,728,594	93.5%	9,237,741,705	99.1%	7,163,671,137	103.8%
65-74	143,303,417	80.8%	240,017,175	81.0%	343,292,965	88.5%	930,432,092	92.5%	8,689,000,890	98.9%	6,262,790,977	105.1%
75-84	45,633,095	118.8%	49,322,782	76.7%	121,205,864	107.0%	506,866,160	116.0%	13,883,785,714	98.6%	1,277,198,335	110.7%
85-99	15,403,219	239.9%	16,508,563	154.3%	25,786,836	139.8%	61,467,096	84.7%	1,297,887,055	100.3%	23,259,787	61.5%

dur_band1 x gender

dur_band1	gender: F		gender: M
dur_band1	Outcome	Ratio	Outcome	Ratio
01	322,942,506	112.1%	946,193,789	96.4%
02	424,637,814	88.4%	1,428,129,267	104.0%
03	697,046,249	100.6%	1,797,111,643	99.8%
04-05	1,788,538,549	94.6%	4,532,906,783	102.3%
06-15	19,652,776,350	100.3%	36,024,713,938	99.8%
16-25	10,371,250,467	100.6%	19,949,489,105	99.7%

dur_band1 x insurance_plan

dur_band1	insurance_plan: Other		insurance_plan: Perm		insurance_plan: Term		insurance_plan: xL
dur_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
01	6,987,296	71.5%	115,387,649	144.0%	816,300,396	95.0%	330,460,954	103.3%
02	12,956,089	93.2%	186,759,670	170.6%	1,193,573,840	98.7%	459,477,482	88.3%
03	10,940,700	62.7%	173,260,241	123.2%	1,539,568,218	97.1%	770,388,733	102.6%
04-05	41,382,619	117.3%	351,202,533	111.1%	3,960,972,351	101.1%	1,967,887,829	95.9%
06-15	92,001,723	94.3%	2,251,458,889	110.9%	24,546,892,971	100.1%	28,787,136,705	99.2%
16-25	23,083,873	172.0%	9,083,854,279	95.8%	6,966,308,277	100.5%	14,247,493,143	102.6%

dur_band1 x ltp

ltp	dur_band1: 01		dur_band1: 02		dur_band1: 03		dur_band1: 04-05		dur_band1: 06-15		dur_band1: 16-25
ltp	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
5 yr	8,474,532	140.6%	5,768,265	70.3%	13,120,000	97.0%	47,280,625	102.7%	258,887,272	99.8%	352,812,820	99.9%
10 yr	250,039,805	87.6%	374,445,553	90.6%	463,240,176	85.9%	1,216,385,768	92.0%	4,556,704,077	103.6%	410,849,344	131.3%
15 yr	106,613,913	95.8%	154,795,986	92.5%	219,264,610	98.1%	520,402,972	92.2%	4,466,179,563	94.8%	550,499,516	229.5%
20 yr	314,198,709	105.4%	447,558,227	106.0%	600,176,032	106.3%	1,543,338,839	108.2%	11,599,090,681	100.9%	3,262,686,119	91.5%
25 yr	3,615,999	48.6%	13,707,800	138.9%	18,921,628	130.4%	49,784,528	143.2%	267,598,048	98.6%	171,594,919	91.7%
30 yr	66,053,409	84.9%	98,748,126	97.2%	130,459,928	98.9%	388,766,012	117.8%	1,983,373,230	102.1%	425,331,080	83.7%
Not Level Term	453,235,980	110.6%	659,812,994	102.4%	954,906,252	105.0%	2,361,669,130	98.2%	31,616,749,475	100.0%	23,567,941,704	99.8%
Unknown	66,903,948	91.4%	97,930,130	113.4%	94,069,266	96.1%	193,817,458	100.1%	928,907,942	95.6%	1,579,024,070	102.7%

dur_band1 x iy_band1

dur_band1	iy_band1: 2010+		iy_band1: 2000-2009		iy_band1: 1990-1999		iy_band1: 1900-1989
dur_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
01	1,269,136,295	100.0%
02	1,852,767,081	100.0%
03	2,494,157,892	100.0%
04-05	4,479,996,408	98.5%	1,841,448,924	103.8%
06-15	1,786,058,616	103.9%	51,370,835,971	99.9%	2,520,595,701	98.9%
16-25			1,833,008,413	98.5%	27,023,551,645	100.1%	1,464,179,514	100.0%

ia_band1 x gender

ia_band1	gender: F		gender: M
ia_band1	Outcome	Ratio	Outcome	Ratio
18-24	210,351,040	88.4%	457,580,041	106.4%
25-34	1,919,168,906	93.6%	4,348,358,909	103.1%
35-44	4,376,241,525	95.6%	12,614,211,612	101.6%
45-54	4,611,772,525	96.3%	16,017,737,790	101.1%
55-64	4,809,017,158	99.7%	14,638,134,932	100.1%
65-74	7,381,394,530	102.0%	9,227,442,986	98.5%
75-84	9,045,279,898	104.3%	6,838,732,052	94.9%
85-99	903,966,353	104.7%	536,346,203	93.0%

ia_band1 x insurance_plan

ia_band1	insurance_plan: Other		insurance_plan: Perm		insurance_plan: Term		insurance_plan: xL
ia_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
18-24	708,665	58.0%	166,230,095	92.6%	267,466,700	103.9%	233,525,621	101.6%
25-34	3,256,794	65.2%	852,327,846	101.0%	4,057,228,956	99.0%	1,354,714,219	102.7%
35-44	21,307,402	115.2%	1,846,365,837	95.1%	11,715,945,894	100.6%	3,406,834,004	100.6%
45-54	57,525,597	130.3%	2,565,363,967	96.2%	12,641,031,673	100.0%	5,365,589,078	101.7%
55-64	47,144,955	78.8%	3,306,501,193	98.1%	8,357,779,097	99.2%	7,735,726,845	101.9%
65-74	35,749,661	100.5%	2,847,699,847	105.8%	1,896,982,781	100.8%	11,828,405,227	98.6%
75-84	21,392,494	100.4%	492,087,214	117.0%	86,369,648	119.9%	15,284,162,594	99.4%
85-99	266,732	14.9%	85,347,262	178.0%	811,304	157.5%	1,353,887,258	97.4%

ia_band1 x ltp

ia_band1	ltp: 5 yr		ltp: 10 yr		ltp: 15 yr		ltp: 20 yr		ltp: 25 yr		ltp: 30 yr		ltp: Not Level Term		ltp: Unknown
ia_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
18-24	17,726,831	84.0%	32,009,383	104.5%	3,343,000	50.1%	80,676,936	114.1%	13,002,500	99.2%	48,824,425	108.2%	403,156,010	97.8%	69,191,996	101.5%
25-34	147,704,951	101.6%	317,080,031	112.2%	150,181,284	111.1%	1,670,129,784	100.3%	164,417,165	95.1%	748,387,532	96.9%	2,274,993,951	101.8%	794,633,117	92.6%
35-44	232,956,798	94.0%	1,049,894,842	107.7%	983,755,234	111.0%	6,112,699,729	99.2%	222,687,940	94.4%	1,620,869,320	100.6%	5,508,205,763	99.0%	1,259,383,511	96.3%
45-54	178,555,603	105.5%	2,312,649,990	105.3%	2,156,659,786	94.8%	6,421,617,312	98.7%	95,336,910	117.1%	652,477,911	101.5%	8,212,889,839	99.9%	599,322,964	111.2%
55-64	76,414,209	100.5%	2,489,338,251	93.6%	2,056,964,977	97.3%	3,341,836,281	103.9%	27,176,407	137.6%	22,072,597	104.0%	11,252,197,601	100.5%	181,151,767	121.3%
65-74	27,015,122	109.9%	1,014,083,731	94.7%	662,777,279	111.1%	140,088,565	96.5%	2,602,000	131.2%	100,000	44.5%	14,725,682,019	99.9%	36,488,800	116.5%
75-84	5,970,000	287.3%	56,608,495	96.8%	4,065,000	99.0%	0	0.0%					15,797,689,100	99.9%	19,679,355	277.5%
85-99			0	0.0%	10,000	7067.2%							1,439,501,252	100.0%	801,304	159.5%

ia_band1 x iy_band1

ia_band1	iy_band1: 1900-1989		iy_band1: 1990-1999		iy_band1: 2000-2009		iy_band1: 2010+
ia_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
18-24	21,957,490	91.9%	248,374,326	89.6%	249,952,870	120.2%	147,646,395	93.0%
25-34	156,424,882	93.9%	2,415,663,468	97.2%	2,774,706,986	99.6%	920,732,479	110.8%
35-44	323,624,196	99.9%	5,875,528,027	94.0%	8,554,094,758	101.8%	2,237,206,156	111.3%
45-54	340,733,447	95.8%	6,605,000,254	97.1%	10,642,985,328	102.1%	3,040,791,286	99.7%
55-64	443,897,049	104.7%	6,912,220,124	103.2%	8,972,404,806	98.8%	3,118,630,111	96.0%
65-74	172,472,214	104.5%	6,127,207,226	105.3%	8,704,756,697	99.4%	1,604,401,379	85.8%
75-84	5,070,236	101.8%	1,325,459,599	112.9%	13,867,379,988	98.4%	686,102,127	112.7%
85-99	0	0.0%	34,694,322	83.1%	1,279,011,875	99.0%	126,606,359	119.4%

gender x insurance_plan

insurance_plan	gender: F		gender: M
insurance_plan	Outcome	Ratio	Outcome	Ratio
Other	76,199,638	107.2%	111,152,662	95.6%
Perm	4,259,680,761	99.0%	7,902,242,500	100.6%
Term	8,746,195,310	96.4%	30,277,420,743	101.1%
xL	20,175,116,226	101.8%	26,387,728,620	98.6%

gender x ltp

ltp	gender: F		gender: M
ltp	Outcome	Ratio	Outcome	Ratio
5 yr	233,647,378	101.9%	452,696,136	99.0%
10 yr	1,294,444,561	102.3%	5,977,220,162	99.5%
15 yr	1,152,920,343	94.6%	4,864,836,217	101.4%
20 yr	4,067,960,544	95.2%	13,699,088,063	101.5%
25 yr	164,969,732	96.4%	360,253,190	101.7%
30 yr	911,017,939	93.9%	2,181,713,846	102.8%
Not Level Term	24,668,222,437	101.3%	34,946,093,098	99.1%
Unknown	764,009,001	96.9%	2,196,643,813	101.1%

gender x iy_band1

iy_band1	gender: F		gender: M
iy_band1	Outcome	Ratio	Outcome	Ratio
1900-1989	415,833,187	97.0%	1,048,346,327	101.2%
1990-1999	10,124,056,124	101.1%	19,420,091,222	99.4%
2000-2009	19,505,796,137	100.2%	35,539,497,171	99.9%
2010+	3,211,506,487	96.2%	8,670,609,805	101.5%

insurance_plan x ltp

ltp	insurance_plan: Term		insurance_plan: Other		insurance_plan: Perm		insurance_plan: xL
ltp	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
5 yr	686,343,514	100.0%
10 yr	7,271,664,723	100.0%
15 yr	6,017,756,560	100.0%
20 yr	17,767,048,607	100.0%
25 yr	525,222,922	100.0%
30 yr	3,092,731,785	100.0%
Not Level Term	702,195,128	100.0%	187,352,300	100.0%	12,161,923,261	100.0%	46,562,844,846	100.0%
Unknown	2,960,652,814	100.0%

insurance_plan x iy_band1

insurance_plan	iy_band1: 1900-1989		iy_band1: 1990-1999		iy_band1: 2000-2009		iy_band1: 2010+
insurance_plan	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
Other	699,086	17.9%	24,035,150	230.6%	95,671,569	96.9%	66,946,495	90.2%
Perm	685,297,227	91.9%	8,414,471,562	96.2%	2,230,633,550	111.2%	831,520,922	125.3%
Term	110,349,436	110.0%	7,507,786,831	99.9%	24,023,250,728	100.4%	7,382,229,058	98.8%
xL	667,833,765	108.7%	13,597,853,803	102.5%	28,695,737,461	98.9%	3,601,419,817	98.1%

ltp x iy_band1

ltp	iy_band1: 1900-1989		iy_band1: 1990-1999		iy_band1: 2000-2009		iy_band1: 2010+
ltp	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
5 yr	11,828,904	87.5%	356,634,986	100.8%	256,413,452	100.6%	61,466,172	96.0%
10 yr	7,313,258	124.1%	395,337,236	127.5%	4,556,557,084	103.2%	2,312,457,145	91.1%
15 yr	770,000	85.8%	786,184,143	141.4%	4,222,555,938	95.8%	1,008,246,479	95.9%
20 yr	27,216,551	136.9%	3,469,245,630	93.0%	11,456,767,739	101.0%	2,813,818,687	105.4%
25 yr	0	0.0%	162,532,838	88.4%	278,353,329	102.4%	84,336,755	121.4%
30 yr	0	0.0%	553,191,034	85.1%	1,883,712,240	103.7%	655,828,511	104.6%
Not Level Term	1,355,860,609	99.2%	22,279,426,292	100.0%	31,476,742,520	99.8%	4,502,286,114	102.1%
Unknown	61,190,192	106.3%	1,541,595,187	104.5%	914,191,006	93.7%	443,676,429	98.3%

Subgroup Variability

This section reproduces Brian Holland’s publication. For background on the tables generated below, please refer to the publication.

## Load custom functions
source("R/functions_BDHGLM.R")

## Generate and format summary tables for each factor column
factor_cols %>%
  map(.f = \(x) {
    ## Call the mainF function to compute weighted averages for GLM factors
    mainF(
      df = ds,
      model = modelGLM,
      rf = x,
      resp = resp_var,
      offset = resp_offset
    ) %>%
      ## Create a flextable from the results
      flextable() %>%
      ## Set header labels for the table
      set_header_labels("rowname" = "") %>%
      ## Format table values as percentages if numeric
      set_formatter(values = function(x) {
        if (is.numeric(x))
          sprintf("%.1f%%", x * 100)
        else
          x
      }) %>%
      ## Set a caption for the table
      set_caption(caption = paste0("Weighted Average GLM Factors for Variable: ", x)) %>%
      ## Set table properties to enable scrolling
      set_table_properties(opts_html = list(
        scroll = list(
          add_css = "max-height: 500px;"
        )
      )) %>%
      autofit() #%>%
      ## Print the flextable
      #knitr::knit_print()
  }) %>%
  ## Set names for each table based on the factor columns
  purrr::set_names(factor_cols) ->
  output_tables 

if(output_format == "html") {
  output_tables %>%
    map(.f=knitr::knit_print) %>%
    ## Generate a tabset from the list of tables
    generate_tabset(
      tabtitle = "Tables of Terms",
      tablevel = 4
    ) %>%
    ## Print the generated tabset
    cat()
} else {
  export_tables_to_excel(
    output_tables,
    paste0(
        exportsRoot,
        "_glm_subgroup_variability.xlsx"
      )
  )
  cat("See included Excel table for additional information.\n")
}

uw

Weighted Average GLM Factors for Variable: uw
	N/1/1	N/2/1	N/2/2	N/3/1	N/3/2	N/3/3	N/4/1	N/4/2	N/4/3	N/4/4	S/1/1	S/2/1	S/2/2	U/1/1
amount_2015vbt	97.2%	78.0%	116.0%	65.3%	72.0%	98.7%	64.1%	76.0%	92.4%	110.2%	111.6%	79.6%	108.8%	139.2%
Factor: uw	100.0%	85.9%	128.2%	72.4%	83.6%	116.4%	71.9%	84.7%	101.1%	123.1%	107.0%	83.6%	111.9%	129.8%
Ave Fac: dur_band1	90.7%	90.4%	90.2%	90.7%	90.1%	90.1%	91.4%	91.1%	91.6%	91.9%	90.7%	91.1%	91.3%	91.2%
Ave Fac: face_amount_band	78.0%	73.3%	74.8%	72.5%	72.1%	72.5%	72.3%	72.6%	72.6%	73.0%	81.4%	75.0%	76.1%	86.4%
Ave Fac: gender	100.7%	100.7%	100.6%	100.7%	100.7%	100.7%	100.8%	100.8%	100.9%	100.8%	100.7%	100.8%	100.7%	100.6%
Ave Fac: ia_band1	91.6%	91.3%	89.8%	92.2%	89.5%	88.1%	93.1%	92.1%	92.1%	91.1%	92.9%	92.9%	92.3%	92.4%
Ave Fac: insurance_plan	88.8%	89.9%	90.3%	83.7%	89.2%	90.9%	78.0%	79.4%	79.5%	82.1%	88.1%	83.5%	85.9%	89.0%
Ave Fac: iy_band1	90.6%	88.8%	88.4%	86.5%	86.3%	85.9%	85.2%	85.4%	84.9%	84.7%	90.8%	86.5%	86.4%	90.6%
Ave Fac: ltp	71.1%	71.1%	71.1%	76.5%	72.8%	71.4%	83.3%	82.4%	82.8%	80.6%	71.6%	78.3%	76.0%	69.5%

face_amount_band

Weighted Average GLM Factors for Variable: face_amount_band
	01 - 0 - 49,999	04 - 50,000 - 99,999	05 - 100,000 - 249,999	06 - 250,000 - 499,999	07 - 500,000 - 999,999	08 - 1,000,000+
amount_2015vbt	132.1%	118.6%	98.9%	88.2%	83.2%	80.4%
Factor: face_amount_band	100.0%	89.3%	79.0%	74.2%	72.5%	70.6%
Ave Fac: dur_band1	90.8%	90.8%	90.8%	90.8%	90.8%	90.5%
Ave Fac: gender	100.5%	100.6%	100.7%	100.7%	100.8%	100.7%
Ave Fac: ia_band1	91.6%	92.1%	92.2%	92.4%	92.4%	89.6%
Ave Fac: insurance_plan	89.5%	90.6%	86.2%	83.3%	83.0%	88.5%
Ave Fac: iy_band1	91.4%	90.5%	88.8%	87.6%	87.1%	86.5%
Ave Fac: ltp	68.6%	70.0%	74.7%	77.2%	77.5%	73.4%
Ave Fac: uw	105.6%	104.0%	99.3%	95.4%	93.2%	95.5%

dur_band1

Weighted Average GLM Factors for Variable: dur_band1
	01	02	03	04-05	06-15	16-25
amount_2015vbt	90.4%	90.8%	87.3%	85.3%	82.4%	98.9%
Factor: dur_band1	100.0%	101.0%	100.0%	95.7%	89.1%	90.9%
Ave Fac: face_amount_band	73.1%	73.1%	73.1%	73.1%	72.8%	77.4%
Ave Fac: gender	100.8%	100.8%	100.8%	100.7%	100.7%	100.7%
Ave Fac: ia_band1	92.6%	92.4%	92.2%	91.9%	90.1%	92.5%
Ave Fac: insurance_plan	81.9%	82.4%	82.7%	83.1%	86.7%	88.0%
Ave Fac: iy_band1	81.4%	81.4%	81.4%	82.4%	86.2%	92.3%
Ave Fac: ltp	79.9%	79.6%	79.3%	79.1%	75.1%	71.5%
Ave Fac: uw	93.4%	92.8%	92.4%	92.4%	95.7%	99.1%

ia_band1

Weighted Average GLM Factors for Variable: ia_band1
	18-24	25-34	35-44	45-54	55-64	65-74	75-84	85-99
amount_2015vbt	107.9%	88.6%	86.7%	86.9%	88.9%	92.8%	82.7%	76.7%
Factor: ia_band1	100.0%	94.0%	95.6%	92.8%	90.4%	92.2%	84.1%	73.8%
Ave Fac: dur_band1	92.4%	91.0%	90.8%	90.9%	91.1%	90.6%	89.5%	89.7%
Ave Fac: face_amount_band	79.2%	74.8%	73.9%	74.5%	75.4%	74.6%	71.8%	71.7%
Ave Fac: gender	100.7%	100.7%	100.8%	100.8%	100.8%	100.6%	100.5%	100.4%
Ave Fac: insurance_plan	85.4%	80.9%	80.5%	81.9%	85.3%	92.5%	95.6%	95.6%
Ave Fac: iy_band1	87.5%	87.9%	87.8%	87.4%	87.4%	87.7%	86.4%	86.0%
Ave Fac: ltp	74.3%	78.3%	78.9%	78.6%	76.2%	69.9%	67.4%	67.3%
Ave Fac: uw	97.1%	92.9%	91.6%	93.0%	95.4%	99.2%	103.1%	108.5%

gender

Weighted Average GLM Factors for Variable: gender
	F	M
amount_2015vbt	87.4%	87.6%
Factor: gender	100.0%	101.1%
Ave Fac: dur_band1	90.5%	90.8%
Ave Fac: face_amount_band	74.8%	73.8%
Ave Fac: ia_band1	89.8%	91.6%
Ave Fac: insurance_plan	89.3%	85.2%
Ave Fac: iy_band1	87.4%	87.3%
Ave Fac: ltp	72.2%	75.8%
Ave Fac: uw	97.5%	95.6%

insurance_plan

Weighted Average GLM Factors for Variable: insurance_plan
	Other	Perm	Term	xL
amount_2015vbt	84.1%	96.4%	83.4%	89.2%
Factor: insurance_plan	100.0%	87.5%	76.0%	95.9%
Ave Fac: dur_band1	93.1%	91.0%	91.0%	90.2%
Ave Fac: face_amount_band	73.0%	79.1%	73.4%	73.7%
Ave Fac: gender	100.7%	100.7%	100.8%	100.6%
Ave Fac: ia_band1	91.2%	92.2%	93.3%	88.7%
Ave Fac: iy_band1	84.5%	91.0%	86.4%	87.4%
Ave Fac: ltp	67.3%	67.3%	84.5%	67.3%
Ave Fac: uw	97.9%	101.3%	90.5%	100.3%

ltp

Weighted Average GLM Factors for Variable: ltp
	5 yr	10 yr	15 yr	20 yr	25 yr	30 yr	Not Level Term	Unknown
amount_2015vbt	117.1%	91.7%	86.2%	78.6%	91.8%	81.0%	90.2%	92.4%
Factor: ltp	100.0%	91.4%	90.8%	81.3%	78.1%	85.1%	67.3%	81.3%
Ave Fac: dur_band1	90.8%	92.2%	90.7%	90.7%	91.0%	91.1%	90.4%	91.5%
Ave Fac: face_amount_band	77.5%	73.1%	73.5%	73.2%	74.8%	72.9%	74.7%	74.6%
Ave Fac: gender	100.7%	100.9%	100.9%	100.8%	100.7%	100.7%	100.6%	100.8%
Ave Fac: ia_band1	93.9%	92.2%	92.3%	93.5%	94.4%	94.6%	89.4%	94.4%
Ave Fac: insurance_plan	76.0%	76.0%	76.0%	76.0%	76.0%	76.0%	94.1%	76.0%
Ave Fac: iy_band1	89.1%	84.5%	85.7%	86.7%	87.4%	86.5%	88.1%	88.5%
Ave Fac: uw	98.4%	93.3%	90.0%	88.9%	103.5%	86.1%	100.3%	97.8%

iy_band1

Weighted Average GLM Factors for Variable: iy_band1
	1900-1989	1990-1999	2000-2009	2010+
amount_2015vbt	116.3%	99.5%	82.7%	85.3%
Factor: iy_band1	100.0%	93.0%	86.2%	81.4%
Ave Fac: dur_band1	90.9%	90.7%	89.4%	95.8%
Ave Fac: face_amount_band	82.0%	77.4%	72.9%	73.2%
Ave Fac: gender	100.8%	100.7%	100.7%	100.8%
Ave Fac: ia_band1	92.8%	92.5%	90.1%	92.1%
Ave Fac: insurance_plan	90.2%	87.8%	86.9%	83.0%
Ave Fac: ltp	68.4%	71.6%	74.8%	79.1%
Ave Fac: uw	101.9%	99.1%	95.9%	92.4%

LightGBM

Data Preparation

First, the data are prepared for LightGBM. LightGBM expects matrices for its inputs. Thereafter, the LightGBM model is trained. Factors are recast as their underlying integer indices.

Model Fitting

The LightGBM model is fit to the training subset using a Poisson objective. The model response is the ratio of response variable and response offset, and the weights are the specified offset. Often, this might be “actual claims” as the response and “expected claims” as the offset.

#==============================================================================#
#### Section 5: lightgbm ####
#==============================================================================#

#### Notes
## Helpful Resources:
## https://lightgbm.readthedocs.io/en/v3.3.2/
## https://christophm.github.io/interpretable-ml-book/shapley.html

## In this section we fit a lightgbm model, an implementation of gradient 
## boosting machines. We also extract interaction feature importance
## This is a way to determine most likely interactions for a linear model
## We also look at Shapley values which can be useful to decompose black-box 
## model predictions

#-----------------------------------------#
##### Fit model and make predictions #####
#-----------------------------------------#

## create lgbm dataset
lgbm.train <- lgb.Dataset(train.x.lgbm, 
                   label  = train.y.lgbm/train.weight.lgbm, 
                   weight = train.weight.lgbm)

lgbm.test <- lgb.Dataset.create.valid(lgbm.train,
                                      test.x.lgbm,
                                      label=test.y.lgbm/test.weight.lgbm)

## define parameters
params <- list(
          objective = "poisson",
          metric = "poisson",
          min_data_in_leaf = 500,
          learning_rate = .3,
          feature_fraction = .75,
          bagging_fraction = 0.50,
          seed = nGBMSeed
)

## train model 
if(bUseCache & file.exists(
  paste0(cacheFileRoot,"_lgb_model.txt")
  ) & !bInvalidateCaches)
{
  lgbm1 <- lgb.load(paste0(cacheFileRoot,"_lgb_model.txt"))
} else {
  lgbm1 <- lgb.train(
           params = params,
           data = lgbm.train,
           nrounds = 2000L#,  ## for demo purposes; switch back to 2000
           #valids=list(test=lgbm.test),
           #early_stopping_rounds = 10
           )
  if(bUseCache)
    lgb.save(lgbm1,paste0(cacheFileRoot,"_lgb_model.txt"))
}

## generate predictions
## note: predictions needed to be multiplied by weights, linear models do this automatically
test[,predictions_lgbm1:=predict(lgbm1, test.x.lgbm) * get(resp_offset)]
train[,predictions_lgbm1:=predict(lgbm1, train.x.lgbm) * get(resp_offset)]

Model Illustrations and Graphics

From this, we can plot decile lift and Lorenz curves.

The decile lift plot can be interpreted as a way to visualize the effectiveness of a predictive model. It divides the data into ten parts (deciles) based on the model’s predictions, from the highest probability of an event occurring to the lowest. The steeper the plot against deciles, the better the segmentation or lift. We see three lines. The “table” line indicates that the expected mortality is relatively constant across these model deciles even though the “actual” mortality and the mortality predicted by the “model” vary substantially, indicating significant risk stratification.

The Lorenz curve describes is another way of visualizing the risk stratification of the model. The more bowed the line is from the y=x axis, the greater the Gini coefficient and the greater the risk stratification.

Understanding the behavior of the interactions as well as gain and cover can give us some macro insight into what the model is doing. The feature interaction table ranks and demonstrates the most important interactions in the model. ‘gain’ refers to the improvement in accuracy brought by a feature to the branches it is on, thus indicating the feature is important. ‘cover’ measures the number of times a feature is used to split the data across all trees regardless of the gain in accuracy achieved. A high gain with a high cover suggests a feature that is very useful across many parts of the dataset.

Lift Curve

#-----------------------------------------#
##### Validation Metrics #####
#-----------------------------------------#

## generate plot
test[,decile.table(get(resp_var),predictions_lgbm1/get(resp_offset),
                   get(resp_offset))] %>%
  pivot_longer(-c(decile,exposures)) %>% 
  as.data.table() %>%
  ggplot(aes(x=decile, y=value, col=name)) +  
  geom_line() +
  scale_x_continuous(breaks=c(1:10)) +
  labs(x="Decile",y="Response")+
  ggtitle("Decile Lift Plot") +
  theme_minimal() +
  scale_y_continuous(labels=scales::comma)

Lorenz Curve

## lorenz plot
test[,lorenz(get(resp_var), predictions_lgbm1 / get(resp_offset), 
             get(resp_offset))]

Feature Importance

The following plot is the feature importance plot which ranks the mean absolute SHAP value for a given feature. It should be noted that being low on the list does not automatically imply that a feature is unimportant. Due to phenomena such as aggregation bias, features with relatively higher numbers of levels can seemingly rank higher than those with lower numbers of levels. Here, the top three tend to have large numbers of levels versus the bottom four.

#-----------------------------------------#
##### Feature Importance #####
#-----------------------------------------#

## get most important features
if(bUseCache & file.exists(
  paste0(cacheFileRoot,"_lgb_imp.rds")
  ) & !bInvalidateCaches) {
  imp <- readRDS(paste0(cacheFileRoot,"_lgb_imp.rds"))
} else {
  imp <- lgb.importance(lgbm1, percentage = TRUE)
  if(bUseCache)
    saveRDS(imp,paste0(cacheFileRoot,"_lgb_imp.rds"))
}

## get most important interactions from EIX library
## warning: very slow
if(bUseCache & file.exists(
  paste0(cacheFileRoot,"_lgb_imp_int.rds")
  ) & !bInvalidateCaches) {
  imp.int <- readRDS(paste0(cacheFileRoot,"_lgb_imp_int.rds"))
} else {
  imp.int <- importance(lgbm1, sm, option = "interactions")
  if(bUseCache)
    saveRDS(imp.int,paste0(cacheFileRoot,"_lgb_imp_int.rds"))
}

#-----------------------------------------#
##### Shap Values #####
#-----------------------------------------#

## get shap values for lightgbm
if(bUseCache & file.exists(
  paste0(cacheFileRoot,"_lgb_shap.rds")
  ) & !bInvalidateCaches) {
  shap_lgbm <- readRDS(paste0(cacheFileRoot,"_lgb_shap.rds"))
} else {
  shap_lgbm <- as.data.table(
    predict(lgbm1, 
            test.x.lgbm, 
            rawscore = FALSE, 
            predcontrib = TRUE) 
    ) %>%
    setnames(names(.),
             c(colnames(test.x.lgbm),"BIAS")
    )
  shap_lgbm[,pred:=exp(Reduce('+',.SD))*test.weight.lgbm] ## reproduce model predictions
  if(bUseCache)
    saveRDS(shap_lgbm,paste0(cacheFileRoot,"_lgb_shap.rds"))
}

set.seed(1337)
if(flgbm_vis_subset < 1 ) {
  shp_int_subset <- sample.int(n=nrow(train),
                             size=nrow(train)*flgbm_vis_subset)
} else {
  shp_int_subset <- 1:nrow(train)
}

if(bUseCache & file.exists(
  paste0(cacheFileRoot,"_lgb_shapviz.rds")
  ) & !bInvalidateCaches)
{
  shp <- readRDS(paste0(cacheFileRoot,"_lgb_shapviz.rds"))
} else {
  shp <- shapviz(
    lgbm1,
    X_pred=train.x.lgbm[shp_int_subset,],
    X=train[shp_int_subset]
  )
  
  setDT(shp$X)
  
  if(bUseCache)
    saveRDS(shp,paste0(cacheFileRoot,"_lgb_shapviz.rds"))
}

## Feature importance
sv_importance(shp) + theme_minimal()

Feature Interaction Table

We also develop a table of interaction strengths, sorted by the total contribution to explaining variation in the data. Again, aggregation bias can distort the ranking, so interpreting the ranking should be taken with caution.

## Convert the 'imp.int' object to a data.table
imp.int <- data.table(imp.int)

## Create a new column 'Feature2' by sorting and collapsing elements of 'Feature'
imp.int[, Feature2 := sapply(Feature, FUN = function(f) {
  paste(sort(unlist(strsplit(f, ":"))), collapse = ":")
})]

## Aggregate 'sumGain', 'sumCover', and 'frequency' by the new 'Feature2' column
imp.int2 <- imp.int[, .(
  sumGain = sum(sumGain),
  sumCover = sum(sumCover),
  frequency = sum(frequency)
), by = .(Feature = Feature2)]

## Calculate additional metrics: meanCover, meanGain, sumGainPct, sumCoverPct
imp.int2[, `:=`(
  meanCover = sumCover / frequency,
  meanGain = sumGain / frequency,
  sumGainPct = sumGain / sum(sumGain),
  sumCoverPct = sumCover / sum(sumCover)
)]

## Split 'Feature' into 'Feature1' and 'Feature2' columns
imp.int2[, c("Feature1", "Feature2") := tstrsplit(Feature, ":")]

## Order by 'sumGain' in descending order and create a flextable with scrollable properties
imp.int2[order(-sumGain)] %>%
  flextable() %>%
  set_table_properties(opts_html = list(
    scroll = list(
      add_css = "max-height: 500px;"
    )
  ))

Feature	sumGain	sumCover	frequency	meanCover	meanGain	sumGainPct	sumCoverPct	Feature1	Feature2
ia_band1:uw	1,228,300,000	22,200,000	3,003	7,393	409,024	0.101945	0.160579	ia_band1	uw
face_amount_band:uw	869,600,000	6,159,000	2,112	2,916	411,742	0.072174	0.044550	face_amount_band	uw
face_amount_band:ia_band1	793,300,000	6,354,000	2,185	2,908	363,066	0.065841	0.045960	face_amount_band	ia_band1
dur_band1:uw	745,800,000	6,395,000	1,677	3,813	444,723	0.061899	0.046257	dur_band1	uw
dur_band1:ia_band1	658,700,000	10,341,000	1,739	5,947	378,781	0.054670	0.074800	dur_band1	ia_band1
dur_band1:face_amount_band	642,300,000	3,509,000	1,522	2,306	422,011	0.053309	0.025382	dur_band1	face_amount_band
dur_band1:ltp	602,600,000	4,825,000	1,101	4,382	547,321	0.050014	0.034901	dur_band1	ltp
ltp:uw	539,000,000	8,995,000	1,432	6,281	376,397	0.044735	0.065064	ltp	uw
ia_band1:ltp	499,900,000	6,590,000	1,294	5,093	386,321	0.041490	0.047667	ia_band1	ltp
insurance_plan:uw	450,500,000	2,941,000	683	4,306	659,590	0.037390	0.021273	insurance_plan	uw
face_amount_band:gender	432,800,000	2,219,600	1,018	2,180	425,147	0.035921	0.016055	face_amount_band	gender
gender:uw	430,400,000	2,691,000	1,119	2,405	384,629	0.035722	0.019465	gender	uw
gender:ia_band1	420,000,000	3,063,000	1,136	2,696	369,718	0.034859	0.022156	gender	ia_band1
face_amount_band:ltp	414,800,000	4,405,000	1,179	3,736	351,824	0.034427	0.031863	face_amount_band	ltp
iy_band1:uw	386,600,000	7,947,000	1,133	7,014	341,218	0.032087	0.057483	iy_band1	uw
ia_band1:iy_band1	384,800,000	6,330,000	1,041	6,081	369,645	0.031937	0.045787	ia_band1	iy_band1
face_amount_band:iy_band1	374,000,000	4,715,000	1,023	4,609	365,591	0.031041	0.034105	face_amount_band	iy_band1
ia_band1:insurance_plan	370,400,000	7,233,000	964	7,503	384,232	0.030742	0.052318	ia_band1	insurance_plan
face_amount_band:insurance_plan	321,900,000	2,630,400	734	3,584	438,556	0.026717	0.019026	face_amount_band	insurance_plan
dur_band1:gender	245,090,000	1,512,500	688	2,198	356,235	0.020342	0.010940	dur_band1	gender
dur_band1:iy_band1	210,720,000	3,843,000	631	6,090	333,946	0.017489	0.027798	dur_band1	iy_band1
dur_band1:insurance_plan	200,320,000	1,770,400	529	3,347	378,677	0.016626	0.012806	dur_band1	insurance_plan
insurance_plan:ltp	186,950,000	1,353,500	164	8,253	1,139,939	0.015516	0.009790	insurance_plan	ltp
gender:insurance_plan	145,750,000	837,300	310	2,701	470,161	0.012097	0.006056	gender	insurance_plan
insurance_plan:iy_band1	133,400,000	3,095,000	427	7,248	312,412	0.011072	0.022387	insurance_plan	iy_band1
iy_band1:ltp	129,660,000	3,644,000	556	6,554	233,201	0.010761	0.026358	iy_band1	ltp
gender:iy_band1	117,310,000	1,447,700	421	3,439	278,646	0.009736	0.010472	gender	iy_band1
gender:ltp	113,770,000	1,203,000	471	2,554	241,550	0.009443	0.008702	gender	ltp

Gain vs. Cover

As noted above, ‘gain’ refers to the improvement in accuracy brought by a feature to the branches it is on, thus indicating the feature is important. ‘cover’ measures the number of times a feature is used to split the data across all trees regardless of the gain in accuracy achieved. A high gain with a high cover suggests a feature that is very useful across many parts of the dataset.

## Create scatter plot
ggplot(imp.int2, aes(x = sumCover, y = sumGain, label = Feature)) + 
  geom_point() + 
  scale_size() +  
  ggrepel::geom_label_repel() +  # Add labels with repulsion
  theme_minimal()

Feature Plots

It is useful to plot SHAP values for their main effects (e.g., SHAP values for face amount band by face amount band) as well as interactions (e.g., same, but stratified in some way by other variables). Traditionally, scatter plots are used. However, due to overplotting, it is not clear what is going on with the SHAP values. Here we use boxplots of the SHAP values instead of scatter plotting. This provides a sense of the spread of the SHAP values along with the median and outliers. This is particularly useful for qualitatively evaluating whether there are any meaningful interactions.

In what follows, red diamonds are mean SHAP values, while blue squares are mean mortality from a subset of the data. Note that SHAP values are partial effects which work in concern with the other features. Therefore, the mean actual mortality will not necessarily be captured by the variability of the feature SHAP values.

## Load external R script
source("R/ilec_shap_plot.R")

## Select top features to plot
featurestoplot <- imp[1:nPlotTopFeatures, Feature]

## Initialize plot list
plist <- list()

## Loop through top features
for (i in 1:nPlotTopFeatures) {
  ## Filter interactions for current feature
  int.vars <- imp.int2[featurestoplot[i] == Feature1 | featurestoplot[i] == Feature2] %>%
    head(nPlotTopInteractions) %>%
    select(Feature1, Feature2) %>%
    pivot_longer(cols = c(Feature1, Feature2), values_to = "Feature") %>%
    distinct() %>%
    filter(Feature != featurestoplot[i])
  
  ## Add SHAP plot to list
  plist <- c(plist, 
             ilec_shap_plot(
               shp,
               featurestoplot[i],
               int.vars$Feature,
               resp_var = resp_var,
               resp_offset = resp_offset,
               train.data = train[shp_int_subset]
             )
  )
}

## Print plots with headers
plist %>%
  iwalk(~ {
    cat('#### ', .y, '\n\n')
    print(.x)
    cat('\n\n')
  })

main effect: uw

uw x ia_band1

uw x face_amount_band

uw x dur_band1

main effect: face_amount_band

face_amount_band x uw

face_amount_band x ia_band1

face_amount_band x dur_band1

main effect: ia_band1

ia_band1 x uw

ia_band1 x face_amount_band

ia_band1 x dur_band1

Some patterns are noticeable. We discuss them for each group. You can visually detect an interaction by checking whether the box plots are all on the same level or not for a given subgroup.

Underwriting (uw)

Main effect
1. The spreads from highest to lowest risk classes are similar across non-smoker class systems.
2. Smoker differentiation is narrower than for 2-class non-smokers.
Interaction with face amount band
1. The interaction between underwriting and face amount band, for the underwriting effect, appears confined mostly to 3-class non-smoker (N/3/*). Higher face amount bands ($250K+ in the light dataset) appear to have larger spread of effects.
Interaction with issue age band: possible narrowing at older ages for 2- or 4-class non-smokers
Interaction with observation year: possible widening of spread of 4-class non-smokers with increasing observation year

Face Amount Band (face_amount_band)

Main effect: expected decrease as face amount band increases
Interaction with underwriting: face amount effect may be interacting with the Unknown smoker category
Interaction with issue age band:
1. Decreasing effect by issue age for lower bands, flipping to increasing effect by issue age for upper issue age bands
2. Put another way, spread of face amount effects decreases with increasing issue age
Interaction with Observation Year: no obvious effect

Issue Age Band (ia_band1)

Main Effect: With the exception of ages 18-24, decreasing issue age effect by issue age
Interaction with face amount band: similar to face amount band, spread decreases with increasing issue age
Interaction with underwriting: substantial changes above issue age 75, qualitatively negligible below age 75
Interaction with observation year: no obvious interaction

Goodness-of-Fit

Unvariate Goodness-of-Fit

## Process each factor column and create formatted tables
map(factor_cols, .f = \(x) {
  ## Convert column name to symbol
  x <- sym(x)
  
  ## Group data by factor column and calculate summary statistics
  train %>%
    group_by(!!x) %>%
    summarize(
      Outcome = sum(amount_actual),
      AM = sum(amount_actual) / sum(predictions_lgbm1)
    ) %>%
    flextable() %>%  # Create a flextable
    set_formatter(  # Format 'AM' as a percentage
      AM = function(x) {
        if (is.numeric(x))
          sprintf("%.1f%%", x * 100)
        else
          x
      }
    ) %>%
    colformat_num(j = "Outcome") %>%  # Format 'Outcome' column
    set_header_labels(  # Set custom header labels
      Outcome = "Outcome",
      AM = "Actual-to-Model"
    ) %>%
    autofit() #%>%
    #knitr::knit_print()  # Print the table in a format suitable for knitting
}) %>%   # Set names for each element in the list
  purrr::set_names(factor_cols) ->
  output_tables

if(output_format == "html") {
  output_tables %>%
    map(.f=knitr::knit_print) %>%
    generate_tabset(  # Generate tabset from the list of tables
      tabtitle = "",
      tablevel = 4
    ) %>%
    cat()  # Print the tabset
} else {
  export_tables_to_excel(
    output_tables,
    paste0(
      exportsRoot,
      "_lightgbm_univariate_goodness_of_fit_tables.xlsx"
    )
  )
  
  cat("See included Excel table for additional information.\n")
  
}

uw

uw	Outcome	Actual-to-Model
N/1/1	19,203,624,009	100.0%
N/2/1	9,661,768,053	100.0%
N/2/2	11,546,531,146	100.0%
N/3/1	5,950,371,727	99.9%
N/3/2	8,810,968,555	100.0%
N/3/3	14,970,350,057	100.0%
N/4/1	6,842,959,121	100.0%
N/4/2	5,162,511,227	100.0%
N/4/3	3,565,224,892	100.0%
N/4/4	3,742,908,023	100.1%
S/1/1	4,365,025,076	100.0%
S/2/1	1,910,521,866	100.1%
S/2/2	1,704,933,729	100.0%
U/1/1	498,038,979	100.1%

face_amount_band

face_amount_band	Outcome	Actual-to-Model
01 - 0 - 49,999	4,193,809,072	100.0%
04 - 50,000 - 99,999	5,283,024,574	100.0%
05 - 100,000 - 249,999	16,241,007,604	100.0%
06 - 250,000 - 499,999	14,745,604,133	100.0%
07 - 500,000 - 999,999	15,180,744,107	100.0%
08 - 1,000,000+	42,291,546,970	100.0%

dur_band1

dur_band1	Outcome	Actual-to-Model
01	1,269,136,295	100.1%
02	1,852,767,081	100.0%
03	2,494,157,892	99.9%
04-05	6,321,445,332	100.1%
06-15	55,677,490,288	100.0%
16-25	30,320,739,572	100.0%

ia_band1

ia_band1	Outcome	Actual-to-Model
18-24	667,931,081	99.9%
25-34	6,267,527,815	100.0%
35-44	16,990,453,137	100.0%
45-54	20,629,510,315	100.0%
55-64	19,447,152,090	100.0%
65-74	16,608,837,516	100.0%
75-84	15,884,011,950	100.0%
85-99	1,440,312,556	100.0%

gender

gender	Outcome	Actual-to-Model
F	33,257,191,935	100.0%
M	64,678,544,525	100.0%

insurance_plan

insurance_plan	Outcome	Actual-to-Model
Other	187,352,300	99.4%
Perm	12,161,923,261	100.0%
Term	39,023,616,053	100.0%
xL	46,562,844,846	100.0%

ltp

ltp	Outcome	Actual-to-Model
5 yr	686,343,514	100.0%
10 yr	7,271,664,723	100.0%
15 yr	6,017,756,560	100.0%
20 yr	17,767,048,607	100.0%
25 yr	525,222,922	100.3%
30 yr	3,092,731,785	100.1%
Not Level Term	59,614,315,535	100.0%
Unknown	2,960,652,814	100.0%

iy_band1

iy_band1	Outcome	Actual-to-Model
1900-1989	1,464,179,514	100.0%
1990-1999	29,544,147,346	100.0%
2000-2009	55,045,293,308	100.0%
2010+	11,882,116,292	100.0%

Bivariate Goodness-of-Fit

## Create a list of unique pairs of factor columns
pairlist <- data.table()
for (i in 1:(length(factor_cols) - 1)) {
  for (j in (i + 1):length(factor_cols)) {
    pairlist <- rbind(pairlist, data.table(F1 = factor_cols[i], F2 = factor_cols[j]))
  }
}

## Generate summary tables and formatted outputs for each pair of factor columns
map2(.x = pairlist$F1, .y = pairlist$F2, .f = \(x, y) {
  ## Convert column names to symbols for tidy evaluation
  xs <- sym(x)
  ys <- sym(y)
  
  ## Choose grouping order based on the number of levels in each factor
  ## Group by the factor with more levels first for better summarization
  if (length(train[, levels(get(x))]) >= length(train[, levels(get(y))])) {
    fttmp <- train %>%
      group_by(!!xs, !!ys) %>%
      summarize(
        Outcome = sum(amount_actual),  # Calculate the total outcome
        Ratio = sprintf("%.1f%%", 100 * sum(amount_actual) / sum(predictions_lgbm1))  # Calculate the actual-to-model ratio
      ) %>%
      pivot_wider(
        names_from = !!ys,  # Pivot the data to widen by the second factor
        values_from = c(Outcome, Ratio),  # Use Outcome and Ratio as values
        names_glue = paste0(y, ": {", y, "}.{.value}"),  # Create new column names using glue syntax
        names_vary = "slowest"  # Handle varying names by the slowest changing variable
      )
  } else {
    fttmp <- train %>%
      group_by(!!ys, !!xs) %>%
      summarize(
        Outcome = sum(amount_actual),  # Calculate the total outcome
        Ratio = sprintf("%.1f%%", 100 * sum(amount_actual) / sum(predictions_lgbm1))  # Calculate the actual-to-model ratio
      ) %>%
      pivot_wider(
        names_from = !!xs,  # Pivot the data to widen by the first factor
        values_from = c(Outcome, Ratio),  # Use Outcome and Ratio as values
        names_glue = paste0(x, ": {", x, "}.{.value}"),  # Create new column names using glue syntax
        names_vary = "slowest"  # Handle varying names by the slowest changing variable
      )
  }
  
  ## Adjust column keys for the flextable
  ## Start with the first column name
  fttmp.colkeys <- names(fttmp)[1]
  ## Add pairs of Outcome and Ratio columns, inserting a blank column between each pair
  for (i in 1:((length(names(fttmp)) - 1) / 2)) {
    fttmp.colkeys <- c(fttmp.colkeys, paste0("blank", i), names(fttmp)[(2 * i):(2 * i + 1)])
  }
  
  ## Create and print the flextable
  fttmp %>%
    flextable(col_keys = fttmp.colkeys) %>%
    ftExtra::span_header(sep = "\\.") %>%
    align(align = 'center', part = "all") %>%
    empty_blanks() %>%
    autofit() #%>%
    #knitr::knit_print()
}) %>%
  ## Set names for each element in the list based on the factor column pairs
  purrr::set_names(pairlist[, paste0(F1, " x ", F2)]) ->
  output_tables

if(output_format == "html") {
  output_tables %>%
    map(.f=knitr::knit_print) %>%
    ## Generate a tabset from the list of formatted tables
    generate_tabset(tabtitle = "", tablevel = 4) %>%
    ## Print the generated tabset
    cat()  
} else {
  export_tables_to_excel(
    output_tables,
    paste0(
      exportsRoot,
      "_lightgbm_bivariate_goodness_of_fit_tables.xlsx"
    )
  )
  
  cat("See included Excel table for additional information.\n")
}

uw x face_amount_band

uw	face_amount_band: 01 - 0 - 49,999		face_amount_band: 04 - 50,000 - 99,999		face_amount_band: 05 - 100,000 - 249,999		face_amount_band: 06 - 250,000 - 499,999		face_amount_band: 07 - 500,000 - 999,999		face_amount_band: 08 - 1,000,000+
uw	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
N/1/1	2,331,609,989	100.1%	2,665,573,911	99.9%	4,377,227,666	100.0%	2,484,222,240	99.9%	2,280,087,941	100.0%	5,064,902,262	100.0%
N/2/1	67,510,912	100.0%	227,491,364	100.0%	1,877,102,924	100.0%	1,581,998,866	100.0%	1,613,477,143	100.2%	4,294,186,844	100.0%
N/2/2	478,620,100	100.0%	716,716,216	100.1%	2,460,749,546	100.1%	1,807,687,840	99.7%	1,536,284,872	100.5%	4,546,472,572	99.9%
N/3/1	4,589,248	106.8%	36,548,841	98.6%	636,379,622	100.1%	1,099,392,788	99.9%	1,269,934,939	100.3%	2,903,526,289	99.8%
N/3/2	15,879,147	95.9%	84,477,290	103.2%	824,647,078	99.4%	1,194,151,445	100.7%	1,294,166,133	99.2%	5,397,647,462	100.1%
N/3/3	89,078,013	100.0%	266,651,094	99.1%	1,685,052,988	100.4%	1,897,715,243	99.7%	1,940,524,074	100.0%	9,091,328,645	100.0%
N/4/1	2,890,726	85.8%	23,168,994	99.2%	533,862,585	99.9%	1,152,655,538	100.3%	1,611,064,623	99.9%	3,519,316,655	99.9%
N/4/2	5,630,152	96.8%	43,592,461	100.0%	580,655,929	99.4%	959,152,506	100.7%	1,142,130,198	99.7%	2,431,349,981	100.0%
N/4/3	2,758,598	104.1%	19,493,024	94.9%	410,982,168	100.6%	652,807,204	99.4%	822,151,439	100.4%	1,657,032,459	100.0%
N/4/4	16,764,842	98.8%	69,558,070	102.0%	501,578,433	100.1%	668,382,546	99.8%	698,077,315	99.8%	1,788,546,817	100.2%
S/1/1	846,343,572	100.0%	864,666,577	100.1%	1,161,486,983	99.6%	510,128,489	100.4%	361,498,469	100.2%	620,900,986	100.1%
S/2/1	21,486,405	97.5%	85,321,233	100.7%	614,066,701	100.0%	400,530,746	100.3%	315,167,594	100.0%	473,949,187	100.1%
S/2/2	74,338,988	100.4%	135,934,467	100.0%	492,786,516	100.4%	294,838,420	100.1%	267,040,410	100.1%	439,994,928	99.2%
U/1/1	236,308,380	99.8%	43,831,032	98.9%	84,428,465	100.6%	41,940,262	99.8%	29,138,957	100.6%	62,391,883	101.1%

uw x dur_band1

uw	dur_band1: 01		dur_band1: 02		dur_band1: 03		dur_band1: 04-05		dur_band1: 06-15		dur_band1: 16-25
uw	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
N/1/1	107,404,206	99.1%	127,611,866	99.4%	186,488,501	100.0%	495,415,170	100.6%	4,962,783,774	99.9%	13,323,920,492	100.0%
N/2/1	38,939,578	98.9%	117,924,754	100.1%	103,706,358	100.7%	187,685,672	99.3%	4,423,311,154	100.2%	4,790,200,537	99.9%
N/2/2	68,820,764	102.6%	89,429,859	99.0%	153,874,204	98.5%	321,513,019	100.8%	6,248,105,591	100.0%	4,664,787,709	100.0%
N/3/1	91,917,216	100.0%	162,393,895	99.7%	239,046,418	100.4%	463,263,819	99.3%	4,110,289,163	100.1%	883,461,216	99.6%
N/3/2	73,068,462	98.2%	142,447,178	101.1%	172,423,532	100.0%	471,435,974	99.7%	7,112,246,310	99.9%	839,347,099	100.7%
N/3/3	209,144,931	100.7%	274,764,174	99.7%	374,385,325	100.5%	1,050,466,173	100.3%	12,379,149,521	99.9%	682,439,933	100.2%
N/4/1	174,316,019	100.9%	228,792,240	99.9%	389,498,840	100.4%	1,034,633,390	99.5%	4,624,074,611	100.1%	391,644,021	99.4%
N/4/2	115,650,082	100.3%	173,110,541	100.1%	196,392,926	99.1%	696,533,900	100.5%	3,717,590,891	100.0%	263,232,887	99.1%
N/4/3	97,264,439	99.8%	192,504,643	101.2%	206,119,588	98.6%	501,287,156	100.6%	2,450,944,955	99.9%	117,104,111	100.4%
N/4/4	153,464,379	99.7%	195,703,696	98.8%	283,447,385	100.5%	660,958,667	100.1%	2,334,500,075	100.3%	114,833,821	96.6%
S/1/1	30,145,775	98.2%	28,847,772	99.1%	40,393,654	98.6%	91,257,938	99.4%	1,121,084,058	99.8%	3,053,295,879	100.1%
S/2/1	42,753,182	99.7%	58,621,979	102.5%	74,856,996	99.8%	177,694,957	99.6%	1,089,203,131	100.0%	467,391,621	100.3%
S/2/2	55,705,006	100.9%	55,852,780	99.5%	68,480,349	99.9%	160,693,564	100.5%	985,575,007	100.0%	378,627,023	99.7%
U/1/1	10,542,256	96.8%	4,761,704	99.2%	5,043,816	98.7%	8,605,933	94.6%	118,632,047	101.2%	350,453,223	100.0%

uw x ia_band1

uw	ia_band1: 18-24		ia_band1: 25-34		ia_band1: 35-44		ia_band1: 45-54		ia_band1: 55-64		ia_band1: 65-74		ia_band1: 75-84		ia_band1: 85-99
uw	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
N/1/1	195,107,686	99.3%	1,468,766,511	99.9%	3,219,766,527	100.1%	3,573,783,086	100.0%	4,457,649,299	100.0%	4,148,815,415	100.0%	1,980,748,076	100.3%	158,987,409	94.0%
N/2/1	87,770,402	102.3%	804,579,965	99.9%	1,676,447,802	99.8%	1,660,839,957	100.1%	1,585,401,151	99.8%	2,098,151,266	100.3%	1,649,244,265	99.4%	99,333,245	112.7%
N/2/2	82,359,855	100.2%	734,969,884	99.5%	1,537,827,224	100.1%	1,935,817,830	99.9%	1,907,603,917	100.5%	2,320,063,956	99.9%	2,742,077,451	100.0%	285,811,029	99.4%
N/3/1	42,971,227	98.3%	542,310,521	100.6%	1,509,846,536	100.2%	1,517,047,115	99.3%	1,083,241,294	100.8%	682,592,797	98.7%	557,962,237	100.5%	14,400,000	101.6%
N/3/2	26,380,775	99.8%	339,627,773	99.6%	1,177,249,350	100.1%	1,552,812,212	99.8%	1,501,263,082	99.3%	1,589,866,999	100.3%	2,421,887,776	100.3%	201,880,588	99.8%
N/3/3	29,150,024	100.1%	391,634,706	99.8%	1,364,327,286	99.6%	1,932,181,705	100.6%	2,306,037,454	99.7%	2,995,484,451	100.0%	5,412,062,505	100.0%	539,471,926	100.1%
N/4/1	26,287,124	97.1%	602,385,596	101.0%	2,097,046,161	99.9%	2,253,038,301	99.8%	1,386,295,039	100.4%	431,665,885	101.7%	43,983,146	78.3%	2,257,869	199.0%
N/4/2	9,905,787	106.8%	233,732,519	99.9%	1,112,629,077	100.1%	1,595,224,194	99.9%	1,413,399,308	100.1%	448,165,240	98.8%	313,744,476	101.6%	35,710,626	97.9%
N/4/3	7,030,001	98.6%	209,711,793	100.1%	705,969,366	99.7%	1,147,207,102	100.7%	986,258,205	99.3%	356,988,649	100.5%	140,290,698	99.1%	11,769,078	103.5%
N/4/4	9,949,832	98.4%	167,099,067	99.2%	638,256,939	100.7%	1,003,674,951	99.7%	958,007,317	100.3%	551,614,283	99.7%	345,232,117	99.8%	69,073,517	101.1%
S/1/1	68,766,581	100.3%	451,737,170	100.2%	1,077,472,651	99.9%	1,226,674,054	99.9%	936,351,195	100.1%	461,642,247	99.8%	139,279,941	102.4%	3,101,237	49.6%
S/2/1	26,876,275	100.6%	179,096,556	99.3%	474,899,912	99.9%	608,218,812	100.3%	402,893,657	99.5%	174,002,125	101.3%	42,414,641	99.5%	2,119,888	284.2%
S/2/2	17,471,313	100.2%	117,414,041	99.5%	333,323,589	100.0%	526,461,474	100.3%	400,538,301	100.1%	233,737,976	99.4%	63,750,567	97.4%	12,236,468	111.0%
U/1/1	37,904,199	98.8%	24,461,713	99.4%	65,390,717	101.3%	96,529,522	99.9%	122,212,871	98.6%	116,046,227	101.2%	31,334,054	102.6%	4,159,676	92.1%

uw x gender

uw	gender: F		gender: M
uw	Outcome	Ratio	Outcome	Ratio
N/1/1	6,871,313,340	100.0%	12,332,310,669	100.0%
N/2/1	3,739,559,716	100.1%	5,922,208,337	100.0%
N/2/2	4,668,956,646	100.0%	6,877,574,500	100.0%
N/3/1	1,982,219,430	100.0%	3,968,152,297	99.9%
N/3/2	3,036,089,596	100.1%	5,774,878,959	99.9%
N/3/3	5,764,067,198	99.8%	9,206,282,859	100.1%
N/4/1	1,822,625,798	100.0%	5,020,333,323	100.0%
N/4/2	997,342,369	100.0%	4,165,168,858	100.0%
N/4/3	561,153,990	100.3%	3,004,070,902	99.9%
N/4/4	894,420,215	99.9%	2,848,487,808	100.1%
S/1/1	1,618,467,456	100.2%	2,746,557,620	99.9%
S/2/1	534,567,286	99.9%	1,375,954,580	100.1%
S/2/2	553,360,607	100.0%	1,151,573,122	99.9%
U/1/1	213,048,288	99.9%	284,990,691	100.2%

uw x insurance_plan

uw	insurance_plan: Other		insurance_plan: Perm		insurance_plan: Term		insurance_plan: xL
uw	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
N/1/1	41,338,662	100.4%	6,000,364,392	100.0%	3,545,626,085	100.0%	9,616,294,870	100.0%
N/2/1	10,081,665	103.9%	1,335,375,364	100.1%	2,232,053,273	99.9%	6,084,257,751	100.1%
N/2/2	15,784,451	94.9%	1,659,464,730	100.0%	3,001,480,499	99.8%	6,869,801,466	100.1%
N/3/1	40,123,942	97.1%	317,915,492	99.5%	3,500,287,255	100.1%	2,092,045,038	99.9%
N/3/2	20,508,375	101.7%	282,009,643	100.0%	3,266,885,153	100.0%	5,241,565,384	100.0%
N/3/3	32,085,711	102.2%	496,907,377	100.1%	4,052,652,325	100.0%	10,388,704,644	100.0%
N/4/1	2,300,000	104.3%	9,330,053	119.4%	6,251,798,169	100.0%	579,530,899	99.3%
N/4/2	7,223,579	96.0%	7,940,683	103.8%	4,382,666,145	100.0%	764,680,820	100.2%
N/4/3	103,684	11.3%	2,495,706	83.6%	3,036,110,841	100.0%	526,514,661	100.0%
N/4/4	3,739,130	125.0%	12,550,388	103.6%	2,742,467,039	100.0%	984,151,466	100.1%
S/1/1	3,801,542	108.0%	1,397,404,613	100.1%	1,062,735,519	99.6%	1,901,083,402	100.1%
S/2/1	6,544,922	101.5%	163,842,775	99.2%	1,131,185,266	100.2%	608,948,903	100.2%
S/2/2	3,059,454	81.3%	189,503,330	100.0%	778,511,625	100.0%	733,859,320	100.0%
U/1/1	657,183	91.0%	286,818,715	100.1%	39,156,859	97.3%	171,406,222	100.6%

uw x ltp

uw	ltp: 5 yr		ltp: 10 yr		ltp: 15 yr		ltp: 20 yr		ltp: 25 yr		ltp: 30 yr		ltp: Not Level Term		ltp: Unknown
uw	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
N/1/1	167,693,145	100.8%	329,259,367	99.4%	313,430,210	100.3%	1,254,368,612	99.9%	22,798,990	101.7%	140,376,344	100.1%	15,709,773,948	100.0%	1,265,923,393	100.0%
N/2/1	109,944,549	99.6%	251,674,560	99.9%	154,929,926	101.5%	1,003,102,210	99.7%	54,969,322	100.7%	91,526,385	101.6%	7,455,632,156	100.1%	539,988,945	99.7%
N/2/2	101,292,185	99.8%	464,466,296	100.6%	319,758,321	98.0%	1,222,999,988	100.1%	275,278,999	101.9%	237,099,784	98.2%	8,573,983,079	100.0%	351,652,494	100.2%
N/3/1	41,020,000	100.9%	402,661,935	99.3%	437,170,279	101.0%	1,876,102,407	99.6%	32,892,500	96.7%	371,739,248	101.5%	2,700,444,573	100.0%	88,340,785	98.1%
N/3/2	62,023,000	99.2%	523,988,195	100.0%	476,579,573	99.5%	1,824,845,270	100.4%	12,110,000	84.6%	210,771,725	98.0%	5,612,039,277	100.0%	88,611,515	102.6%
N/3/3	59,980,899	97.5%	896,107,443	100.4%	652,611,790	100.4%	1,842,591,351	100.3%	16,475,000	93.1%	201,629,249	99.2%	11,161,374,076	100.0%	139,580,249	98.7%
N/4/1	4,410,000	130.9%	1,142,209,199	99.8%	1,124,192,978	99.7%	3,080,000,938	99.9%	36,372,000	101.2%	845,189,202	100.8%	591,162,804	99.6%	19,422,000	108.1%
N/4/2	985,000	72.4%	899,393,774	100.3%	918,062,758	100.1%	2,118,167,322	99.9%	14,432,100	92.3%	414,673,900	99.7%	779,845,082	100.2%	16,951,291	96.6%
N/4/3	300,000	74.6%	711,850,279	99.6%	663,787,225	100.1%	1,348,193,434	99.9%	8,775,000	136.9%	270,343,655	100.8%	529,114,051	99.8%	32,861,248	102.7%
N/4/4	1,900,000	140.0%	861,440,026	99.9%	549,878,289	99.9%	1,117,119,304	100.2%	8,832,928	90.4%	186,021,492	99.5%	1,000,440,984	100.2%	17,275,000	100.8%
S/1/1	77,649,707	99.6%	220,477,191	101.2%	92,140,018	97.7%	325,974,726	100.1%	37,348,584	99.0%	43,126,718	94.3%	3,324,217,468	100.1%	244,090,664	99.9%
S/2/1	35,606,411	101.5%	312,687,204	99.6%	190,844,063	100.8%	450,793,441	99.4%	2,925,000	122.6%	50,493,377	103.5%	784,528,034	100.0%	82,644,336	101.0%
S/2/2	19,448,975	97.3%	251,942,737	99.9%	122,991,130	100.9%	294,652,404	100.4%	1,050,000	64.9%	28,434,706	101.5%	932,515,030	99.7%	53,898,747	100.6%
U/1/1	4,089,643	104.6%	3,506,517	89.2%	1,380,000	54.0%	8,137,200	120.8%	962,499	80.5%	1,306,000	78.4%	459,244,973	100.3%	19,412,147	97.5%

uw x iy_band1

uw	iy_band1: 1900-1989		iy_band1: 1990-1999		iy_band1: 2000-2009		iy_band1: 2010+
uw	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
N/1/1	982,160,538	99.7%	12,534,732,161	100.0%	4,842,664,276	100.0%	844,067,034	99.9%
N/2/1	112,802,132	103.1%	4,655,974,027	99.9%	4,443,736,468	100.1%	449,255,426	100.3%
N/2/2	107,232,938	102.8%	4,512,424,274	100.0%	6,312,557,239	100.0%	614,316,695	99.3%
N/3/1	282,000	31.1%	1,044,203,844	99.8%	3,887,317,728	99.8%	1,018,568,155	100.6%
N/3/2	250,000	17.7%	1,008,204,301	100.5%	6,934,988,157	100.0%	867,526,097	99.5%
N/3/3	4,055,561	57.8%	755,727,777	100.2%	12,308,187,396	100.0%	1,902,379,323	99.9%
N/4/1	0	0.0%	394,664,512	100.2%	4,606,160,597	99.9%	1,842,134,012	100.0%
N/4/2			308,834,925	99.5%	3,702,484,884	100.0%	1,151,191,418	100.1%
N/4/3			158,854,990	100.9%	2,386,096,189	99.8%	1,020,273,713	100.3%
N/4/4	0	0.0%	146,243,472	99.0%	2,314,552,528	100.3%	1,282,112,023	99.8%
S/1/1	212,607,764	100.1%	2,836,569,692	100.1%	1,124,908,213	100.0%	190,939,407	98.9%
S/2/1	7,889,930	88.0%	479,056,419	100.4%	1,077,182,521	99.8%	346,392,996	100.7%
S/2/2	7,665,540	106.5%	388,469,845	99.3%	986,492,416	100.1%	322,305,928	100.1%
U/1/1	29,233,111	104.7%	320,187,107	99.7%	117,964,696	100.7%	30,654,065	97.4%

face_amount_band x dur_band1

face_amount_band	dur_band1: 01		dur_band1: 02		dur_band1: 03		dur_band1: 04-05		dur_band1: 06-15		dur_band1: 16-25
face_amount_band	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
01 - 0 - 49,999	37,859,834	99.5%	46,746,236	99.3%	59,349,580	99.5%	129,333,013	100.4%	856,334,359	100.1%	3,064,186,050	100.0%
04 - 50,000 - 99,999	52,511,061	99.2%	66,787,618	100.7%	90,269,149	100.2%	206,424,919	99.9%	1,469,057,939	99.9%	3,397,973,888	100.0%
05 - 100,000 - 249,999	191,101,070	100.1%	269,652,985	100.1%	371,980,209	99.7%	855,544,316	100.2%	7,074,965,104	100.0%	7,477,763,920	100.0%
06 - 250,000 - 499,999	209,469,031	100.4%	304,453,835	100.2%	416,690,820	100.1%	1,000,438,105	99.9%	8,109,440,540	100.0%	4,705,111,802	100.1%
07 - 500,000 - 999,999	215,479,316	100.0%	350,555,118	100.2%	461,397,659	100.0%	1,167,345,786	100.2%	8,791,983,362	100.0%	4,193,982,866	100.0%
08 - 1,000,000+	562,715,983	100.1%	814,571,289	99.7%	1,094,470,475	100.0%	2,962,359,193	100.0%	29,375,708,984	100.0%	7,481,721,046	99.9%

face_amount_band x ia_band1

ia_band1	face_amount_band: 01 - 0 - 49,999		face_amount_band: 04 - 50,000 - 99,999		face_amount_band: 05 - 100,000 - 249,999		face_amount_band: 06 - 250,000 - 499,999		face_amount_band: 07 - 500,000 - 999,999		face_amount_band: 08 - 1,000,000+
ia_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
18-24	75,552,784	99.8%	133,231,819	100.0%	220,299,419	99.6%	115,618,404	100.1%	61,754,428	100.3%	61,474,227	100.3%
25-34	192,799,300	100.6%	458,412,108	99.7%	1,476,938,558	100.2%	1,398,298,659	99.7%	1,378,321,244	100.0%	1,362,757,946	100.0%
35-44	421,318,794	99.6%	898,534,810	100.2%	3,004,123,357	100.0%	3,258,823,688	100.1%	3,627,163,035	99.9%	5,780,489,453	100.0%
45-54	861,129,196	100.0%	1,234,583,452	100.0%	4,027,597,384	100.0%	3,776,175,782	100.0%	3,750,396,726	99.9%	6,979,627,775	100.0%
55-64	1,371,789,823	100.0%	1,302,388,605	100.0%	4,041,501,912	99.9%	3,254,795,121	100.1%	2,876,205,825	100.1%	6,600,470,804	99.9%
65-74	1,035,580,594	100.0%	950,356,588	99.9%	2,489,528,859	100.1%	1,929,056,738	100.0%	2,107,668,820	100.2%	8,096,645,917	100.0%
75-84	221,626,122	100.5%	285,327,929	99.8%	901,837,477	100.1%	915,794,848	99.9%	1,257,117,333	100.0%	12,302,308,241	100.0%
85-99	14,012,459	94.2%	20,189,263	101.4%	79,180,638	99.3%	97,040,893	100.9%	122,116,696	100.2%	1,107,772,607	100.0%

face_amount_band x gender

face_amount_band	gender: F		gender: M
face_amount_band	Outcome	Ratio	Outcome	Ratio
01 - 0 - 49,999	2,082,160,405	100.0%	2,111,648,667	100.0%
04 - 50,000 - 99,999	2,075,208,647	100.1%	3,207,815,927	99.9%
05 - 100,000 - 249,999	5,592,847,725	100.0%	10,648,159,879	100.0%
06 - 250,000 - 499,999	4,715,664,348	99.9%	10,029,939,785	100.1%
07 - 500,000 - 999,999	4,435,784,376	100.2%	10,744,959,731	99.9%
08 - 1,000,000+	14,355,526,434	100.0%	27,936,020,536	100.0%

face_amount_band x insurance_plan

face_amount_band	insurance_plan: Other		insurance_plan: Perm		insurance_plan: Term		insurance_plan: xL
face_amount_band	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
01 - 0 - 49,999	4,987,841	90.2%	2,341,324,220	99.9%	430,107,626	100.2%	1,417,389,385	100.1%
04 - 50,000 - 99,999	5,666,783	100.5%	1,269,348,289	100.1%	1,013,673,047	99.8%	2,994,336,455	100.0%
05 - 100,000 - 249,999	34,041,514	99.7%	2,315,213,211	100.0%	6,874,173,321	100.0%	7,017,579,558	100.0%
06 - 250,000 - 499,999	28,639,038	99.5%	1,510,214,779	100.0%	8,348,184,153	100.1%	4,858,566,163	99.9%
07 - 500,000 - 999,999	18,132,026	98.8%	1,434,005,349	100.2%	8,575,363,506	99.8%	5,153,243,226	100.2%
08 - 1,000,000+	95,885,098	99.9%	3,291,817,413	100.0%	13,782,114,400	100.0%	25,121,730,059	100.0%

face_amount_band x ltp

ltp	face_amount_band: 01 - 0 - 49,999		face_amount_band: 04 - 50,000 - 99,999		face_amount_band: 05 - 100,000 - 249,999		face_amount_band: 06 - 250,000 - 499,999		face_amount_band: 07 - 500,000 - 999,999		face_amount_band: 08 - 1,000,000+
ltp	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
5 yr	40,106,609	101.5%	108,758,076	99.7%	223,852,354	99.8%	134,040,438	100.0%	88,356,037	100.1%	91,230,000	100.3%
10 yr	79,471,212	100.1%	152,307,148	100.5%	1,212,386,424	100.0%	1,312,587,882	99.9%	1,417,042,517	99.9%	3,097,869,540	100.0%
15 yr	63,154,393	99.8%	171,454,282	99.3%	1,115,915,003	100.1%	1,284,208,527	100.2%	1,254,669,304	99.8%	2,128,355,051	99.9%
20 yr	113,505,987	100.6%	278,179,336	99.8%	2,893,988,498	99.9%	3,894,785,171	100.1%	4,168,688,693	99.9%	6,417,900,922	100.0%
25 yr	9,580,391	103.9%	26,792,196	100.0%	154,206,000	100.5%	162,940,708	100.3%	101,887,627	100.5%	69,816,000	98.7%
30 yr	6,329,793	102.1%	31,267,129	96.9%	436,286,383	100.6%	721,667,111	99.6%	809,282,434	100.1%	1,087,898,935	100.3%
Not Level Term	3,802,855,700	100.0%	4,317,239,186	100.0%	9,477,900,932	100.0%	6,615,106,546	100.0%	6,758,480,601	100.1%	28,642,732,570	100.0%
Unknown	78,804,987	99.8%	197,027,221	99.9%	726,472,010	100.4%	620,267,750	99.9%	582,336,894	99.8%	755,743,952	99.9%

face_amount_band x iy_band1

face_amount_band	iy_band1: 1900-1989		iy_band1: 1990-1999		iy_band1: 2000-2009		iy_band1: 2010+
face_amount_band	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
01 - 0 - 49,999	267,683,109	99.9%	2,808,814,596	100.0%	833,122,239	100.1%	284,189,128	99.9%
04 - 50,000 - 99,999	267,216,839	100.1%	3,149,056,719	99.9%	1,444,089,389	100.1%	422,661,627	100.0%
05 - 100,000 - 249,999	404,345,538	99.7%	7,205,856,453	100.0%	6,896,526,088	100.0%	1,734,279,525	100.0%
06 - 250,000 - 499,999	158,023,865	100.0%	4,708,333,873	100.1%	7,919,827,744	100.0%	1,959,418,651	99.9%
07 - 500,000 - 999,999	136,967,257	99.5%	4,219,050,921	100.1%	8,611,625,509	99.9%	2,213,100,420	100.3%
08 - 1,000,000+	229,942,906	100.5%	7,453,034,784	99.9%	29,340,102,339	100.0%	5,268,466,941	99.9%

dur_band1 x ia_band1

ia_band1	dur_band1: 01		dur_band1: 02		dur_band1: 03		dur_band1: 04-05		dur_band1: 06-15		dur_band1: 16-25
ia_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
18-24	38,014,843	100.7%	30,257,763	100.6%	27,602,930	98.2%	48,778,652	99.3%	247,658,415	100.1%	275,618,478	99.9%
25-34	134,511,109	99.4%	163,517,330	99.4%	194,704,716	100.9%	424,147,771	100.0%	2,807,372,903	100.0%	2,543,273,986	99.9%
35-44	290,877,481	100.1%	372,345,417	100.5%	502,388,214	99.8%	1,120,149,604	100.2%	8,625,261,989	100.0%	6,079,430,432	100.0%
45-54	284,754,759	100.3%	443,414,694	99.4%	668,187,442	99.7%	1,648,875,363	100.2%	10,888,781,617	100.1%	6,695,496,440	99.9%
55-64	316,638,372	100.2%	537,383,357	100.0%	610,988,925	100.0%	1,580,728,594	100.0%	9,237,741,705	99.8%	7,163,671,137	100.2%
65-74	143,303,417	99.5%	240,017,175	100.9%	343,292,965	100.4%	930,432,092	99.4%	8,689,000,890	100.1%	6,262,790,977	100.0%
75-84	45,633,095	99.5%	49,322,782	97.8%	121,205,864	100.0%	506,866,160	100.7%	13,883,785,714	99.9%	1,277,198,335	100.5%
85-99	15,403,219	106.1%	16,508,563	98.1%	25,786,836	97.1%	61,467,096	98.8%	1,297,887,055	100.9%	23,259,787	69.4%

dur_band1 x gender

dur_band1	gender: F		gender: M
dur_band1	Outcome	Ratio	Outcome	Ratio
01	322,942,506	99.9%	946,193,789	100.1%
02	424,637,814	100.3%	1,428,129,267	99.8%
03	697,046,249	100.3%	1,797,111,643	99.8%
04-05	1,788,538,549	100.0%	4,532,906,783	100.1%
06-15	19,652,776,350	100.0%	36,024,713,938	100.0%
16-25	10,371,250,467	100.0%	19,949,489,105	100.0%

dur_band1 x insurance_plan

dur_band1	insurance_plan: Other		insurance_plan: Perm		insurance_plan: Term		insurance_plan: xL
dur_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
01	6,987,296	100.6%	115,387,649	99.4%	816,300,396	100.2%	330,460,954	100.1%
02	12,956,089	100.9%	186,759,670	99.8%	1,193,573,840	100.0%	459,477,482	99.9%
03	10,940,700	97.9%	173,260,241	100.2%	1,539,568,218	100.0%	770,388,733	99.8%
04-05	41,382,619	100.8%	351,202,533	99.6%	3,960,972,351	100.0%	1,967,887,829	100.2%
06-15	92,001,723	98.7%	2,251,458,889	100.1%	24,546,892,971	100.0%	28,787,136,705	100.0%
16-25	23,083,873	99.3%	9,083,854,279	100.0%	6,966,308,277	99.9%	14,247,493,143	100.0%

dur_band1 x ltp

ltp	dur_band1: 01		dur_band1: 02		dur_band1: 03		dur_band1: 04-05		dur_band1: 06-15		dur_band1: 16-25
ltp	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
5 yr	8,474,532	101.0%	5,768,265	89.9%	13,120,000	99.3%	47,280,625	100.7%	258,887,272	100.3%	352,812,820	99.9%
10 yr	250,039,805	100.0%	374,445,553	100.4%	463,240,176	99.7%	1,216,385,768	100.0%	4,556,704,077	100.0%	410,849,344	99.6%
15 yr	106,613,913	99.4%	154,795,986	100.0%	219,264,610	100.9%	520,402,972	99.3%	4,466,179,563	100.0%	550,499,516	100.2%
20 yr	314,198,709	100.6%	447,558,227	100.0%	600,176,032	99.4%	1,543,338,839	100.2%	11,599,090,681	100.0%	3,262,686,119	99.9%
25 yr	3,615,999	76.5%	13,707,800	112.5%	18,921,628	102.6%	49,784,528	99.8%	267,598,048	99.8%	171,594,919	100.6%
30 yr	66,053,409	101.4%	98,748,126	97.3%	130,459,928	101.9%	388,766,012	100.3%	1,983,373,230	100.1%	425,331,080	100.1%
Not Level Term	453,235,980	99.9%	659,812,994	99.9%	954,906,252	99.9%	2,361,669,130	100.1%	31,616,749,475	100.0%	23,567,941,704	100.0%
Unknown	66,903,948	100.2%	97,930,130	100.1%	94,069,266	100.3%	193,817,458	100.3%	928,907,942	100.2%	1,579,024,070	99.8%

dur_band1 x iy_band1

dur_band1	iy_band1: 2010+		iy_band1: 2000-2009		iy_band1: 1990-1999		iy_band1: 1900-1989
dur_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
01	1,269,136,295	100.1%
02	1,852,767,081	100.0%
03	2,494,157,892	99.9%
04-05	4,479,996,408	100.1%	1,841,448,924	99.9%
06-15	1,786,058,616	99.7%	51,370,835,971	100.0%	2,520,595,701	99.9%
16-25			1,833,008,413	99.7%	27,023,551,645	100.0%	1,464,179,514	100.0%

ia_band1 x gender

ia_band1	gender: F		gender: M
ia_band1	Outcome	Ratio	Outcome	Ratio
18-24	210,351,040	100.2%	457,580,041	99.8%
25-34	1,919,168,906	99.9%	4,348,358,909	100.0%
35-44	4,376,241,525	100.0%	12,614,211,612	100.0%
45-54	4,611,772,525	100.0%	16,017,737,790	100.0%
55-64	4,809,017,158	100.1%	14,638,134,932	99.9%
65-74	7,381,394,530	99.9%	9,227,442,986	100.1%
75-84	9,045,279,898	100.0%	6,838,732,052	100.0%
85-99	903,966,353	100.2%	536,346,203	99.8%

ia_band1 x insurance_plan

ia_band1	insurance_plan: Other		insurance_plan: Perm		insurance_plan: Term		insurance_plan: xL
ia_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
18-24	708,665	69.0%	166,230,095	99.8%	267,466,700	100.2%	233,525,621	99.8%
25-34	3,256,794	74.4%	852,327,846	100.2%	4,057,228,956	99.9%	1,354,714,219	100.1%
35-44	21,307,402	99.6%	1,846,365,837	100.1%	11,715,945,894	100.0%	3,406,834,004	99.8%
45-54	57,525,597	103.6%	2,565,363,967	99.8%	12,641,031,673	100.0%	5,365,589,078	100.2%
55-64	47,144,955	97.2%	3,306,501,193	100.2%	8,357,779,097	100.0%	7,735,726,845	99.9%
65-74	35,749,661	102.3%	2,847,699,847	99.9%	1,896,982,781	99.9%	11,828,405,227	100.1%
75-84	21,392,494	100.5%	492,087,214	99.4%	86,369,648	101.2%	15,284,162,594	100.0%
85-99	266,732	19.4%	85,347,262	105.1%	811,304	75.2%	1,353,887,258	99.8%

ia_band1 x ltp

ia_band1	ltp: 5 yr		ltp: 10 yr		ltp: 15 yr		ltp: 20 yr		ltp: 25 yr		ltp: 30 yr		ltp: Not Level Term		ltp: Unknown
ia_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
18-24	17,726,831	97.1%	32,009,383	100.5%	3,343,000	83.0%	80,676,936	101.0%	13,002,500	96.2%	48,824,425	99.6%	403,156,010	99.9%	69,191,996	101.5%
25-34	147,704,951	100.5%	317,080,031	99.8%	150,181,284	99.2%	1,670,129,784	100.1%	164,417,165	99.9%	748,387,532	99.8%	2,274,993,951	100.0%	794,633,117	99.9%
35-44	232,956,798	99.9%	1,049,894,842	99.9%	983,755,234	99.9%	6,112,699,729	100.1%	222,687,940	99.1%	1,620,869,320	100.1%	5,508,205,763	100.0%	1,259,383,511	100.0%
45-54	178,555,603	100.2%	2,312,649,990	99.9%	2,156,659,786	100.1%	6,421,617,312	100.0%	95,336,910	101.6%	652,477,911	100.3%	8,212,889,839	100.0%	599,322,964	99.9%
55-64	76,414,209	99.1%	2,489,338,251	100.0%	2,056,964,977	100.0%	3,341,836,281	99.9%	27,176,407	109.8%	22,072,597	104.3%	11,252,197,601	100.0%	181,151,767	101.6%
65-74	27,015,122	90.2%	1,014,083,731	100.2%	662,777,279	100.2%	140,088,565	98.4%	2,602,000	106.3%	100,000	43.9%	14,725,682,019	100.0%	36,488,800	93.5%
75-84	5,970,000	241.6%	56,608,495	97.6%	4,065,000	71.7%	0	0.0%					15,797,689,100	100.0%	19,679,355	104.3%
85-99			0	0.0%	10,000	9460.3%							1,439,501,252	100.0%	801,304	75.5%

ia_band1 x iy_band1

ia_band1	iy_band1: 1900-1989		iy_band1: 1990-1999		iy_band1: 2000-2009		iy_band1: 2010+
ia_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
18-24	21,957,490	100.8%	248,374,326	99.9%	249,952,870	99.9%	147,646,395	100.1%
25-34	156,424,882	99.6%	2,415,663,468	100.0%	2,774,706,986	100.0%	920,732,479	99.8%
35-44	323,624,196	100.6%	5,875,528,027	99.9%	8,554,094,758	100.0%	2,237,206,156	100.0%
45-54	340,733,447	98.7%	6,605,000,254	99.9%	10,642,985,328	100.1%	3,040,791,286	100.0%
55-64	443,897,049	100.4%	6,912,220,124	100.1%	8,972,404,806	99.9%	3,118,630,111	99.9%
65-74	172,472,214	99.7%	6,127,207,226	100.1%	8,704,756,697	99.9%	1,604,401,379	100.2%
75-84	5,070,236	132.1%	1,325,459,599	100.2%	13,867,379,988	100.0%	686,102,127	100.1%
85-99	0	0.0%	34,694,322	85.6%	1,279,011,875	100.8%	126,606,359	97.3%

gender x insurance_plan

insurance_plan	gender: F		gender: M
insurance_plan	Outcome	Ratio	Outcome	Ratio
Other	76,199,638	98.3%	111,152,662	100.2%
Perm	4,259,680,761	100.1%	7,902,242,500	100.0%
Term	8,746,195,310	100.1%	30,277,420,743	100.0%
xL	20,175,116,226	100.0%	26,387,728,620	100.0%

gender x ltp

ltp	gender: F		gender: M
ltp	Outcome	Ratio	Outcome	Ratio
5 yr	233,647,378	99.9%	452,696,136	100.1%
10 yr	1,294,444,561	100.2%	5,977,220,162	99.9%
15 yr	1,152,920,343	99.7%	4,864,836,217	100.0%
20 yr	4,067,960,544	100.1%	13,699,088,063	100.0%
25 yr	164,969,732	100.1%	360,253,190	100.3%
30 yr	911,017,939	100.0%	2,181,713,846	100.1%
Not Level Term	24,668,222,437	100.0%	34,946,093,098	100.0%
Unknown	764,009,001	100.2%	2,196,643,813	99.9%

gender x iy_band1

iy_band1	gender: F		gender: M
iy_band1	Outcome	Ratio	Outcome	Ratio
1900-1989	415,833,187	100.0%	1,048,346,327	100.0%
1990-1999	10,124,056,124	100.0%	19,420,091,222	100.0%
2000-2009	19,505,796,137	100.0%	35,539,497,171	100.0%
2010+	3,211,506,487	100.0%	8,670,609,805	100.0%

insurance_plan x ltp

ltp	insurance_plan: Term		insurance_plan: Other		insurance_plan: Perm		insurance_plan: xL
ltp	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
5 yr	686,343,514	100.0%
10 yr	7,271,664,723	100.0%
15 yr	6,017,756,560	100.0%
20 yr	17,767,048,607	100.0%
25 yr	525,222,922	100.3%
30 yr	3,092,731,785	100.1%
Not Level Term	702,195,128	99.0%	187,352,300	99.4%	12,161,923,261	100.0%	46,562,844,846	100.0%
Unknown	2,960,652,814	100.0%

insurance_plan x iy_band1

insurance_plan	iy_band1: 1900-1989		iy_band1: 1990-1999		iy_band1: 2000-2009		iy_band1: 2010+
insurance_plan	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
Other	699,086	11.5%	24,035,150	129.3%	95,671,569	99.6%	66,946,495	98.9%
Perm	685,297,227	100.6%	8,414,471,562	100.0%	2,230,633,550	100.0%	831,520,922	99.8%
Term	110,349,436	98.8%	7,507,786,831	100.0%	24,023,250,728	100.0%	7,382,229,058	100.0%
xL	667,833,765	100.3%	13,597,853,803	100.0%	28,695,737,461	100.0%	3,601,419,817	100.0%

ltp x iy_band1

ltp	iy_band1: 1900-1989		iy_band1: 1990-1999		iy_band1: 2000-2009		iy_band1: 2010+
ltp	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
5 yr	11,828,904	91.8%	356,634,986	100.4%	256,413,452	100.3%	61,466,172	98.7%
10 yr	7,313,258	93.4%	395,337,236	100.0%	4,556,557,084	100.0%	2,312,457,145	100.0%
15 yr	770,000	52.6%	786,184,143	99.9%	4,222,555,938	100.0%	1,008,246,479	100.0%
20 yr	27,216,551	110.5%	3,469,245,630	100.0%	11,456,767,739	100.0%	2,813,818,687	99.9%
25 yr	0	0.0%	162,532,838	99.5%	278,353,329	101.1%	84,336,755	99.1%
30 yr	0	0.0%	553,191,034	100.5%	1,883,712,240	99.9%	655,828,511	100.4%
Not Level Term	1,355,860,609	100.0%	22,279,426,292	100.0%	31,476,742,520	100.0%	4,502,286,114	100.0%
Unknown	61,190,192	98.3%	1,541,595,187	100.0%	914,191,006	100.1%	443,676,429	100.1%

Elastic Net GLM

Background

Elastic net regularization allows the modeler to combine both LASSO and ridge penalties into a single model.

As one may recall, ordinary least squares regression requires minimizing the squared difference of the response variable and the predicted values. In symbols,

\[ \underset{\beta}{\arg\min} \sum_{n}(y-X\beta)^{2} \]

This is equivalent to maximum likelihood estimation, where one assumes that the response variable $y$ is normally distributed with mean $X\beta$ and variance $\sigma^{2}I_{k x k}$. The maximum is taken with respect to $\beta$, and the variance parameter is assumed to be fixed but unknown.

The LASSO and ridge regression methods each add an additional penalty term on the coefficients $\beta$. The LASSO adds the sum of the absolute values of the parameters $\beta$ subject to a tunable weight, $\lambda$. This term incentivizes the fitting algorithm to fit toward parameter values close to 0.

\[ \underset{\beta}{\arg\min} \sum_{n}(y-X\beta)^{2}+ \lambda\sum_{k}|\beta_{k}|\]

The ridge penalty adds the sum of the squares of the parameters $\beta$, subject to a tunable weight, $\alpha$. This term also incentivizes the fitting algorithm to fit toward parameter values close to 0.

\[ \underset{\beta}{\arg\min} \sum_{n}(y-X\beta)^{2}+ \alpha\sum_{k}\beta_{k}^{2}\]

What may be new to some readers is that in both cases, for special $\lambda$ or $\alpha$, the minimizers of these expressions correspond to the Bayesian maximum a posteriori (MAP) estimators for specific prior distributions for $\beta$. In the ridge case, the prior is the normal distribution with mean 0 and covariance $\tau^{2} I_{k \times k}$ for some assumed $\tau^{2}$.

For the LASSO, the prior is the double-exponential or Laplace distribution with mean 0 and parameter $\tau$.

In either case, it can be shown that if $\sigma^{2}$ and $\tau$ are known, the penalizing weights have unique solutions and are equivalent to the $k$ term in Bühlmann credibility. In practice, the penalizing weights are unknown and must be tuned. The resulting optimal $\beta$ is also credible from a Bayesian perspective. Moreover, it can be shown that these facts carry over to the GLM case.

Data Preparation

Elastic net GLMs as implemented in the glmnet package require that the inputs be converted to model matrices.

Model Fitting

Once the data are set up, we can calibrate a LASSO penalty, lambda, using n-fold cross-validation.

## In this section we fit a penalized linear regression model using glmnet. 
## We first fine-tune hyperparameter alpha using cross-validation
## Then we refit the model using the optimal alpha

#-----------------------------------------#
##### Perform Cross-Validation #####
#-----------------------------------------#
## Load or fit the cross-validated glmnet model

## Check if the cached model exists and is valid
if (bUseCache & file.exists(paste0(cacheFileRoot, "_glmnet_int_cv_model.rds")) & !bInvalidateCaches) {
  ## Load the cached model if it exists
  cvfit <- readRDS(paste0(cacheFileRoot, "_glmnet_int_cv_model.rds"))
} else {
  ## Initialize a cluster for parallel processing
  cl <- makeCluster(nGLMNetCores)
  registerDoParallel(cl)
  
  ## Set glmnet control options, enable iteration trace if debugging
  glmnet.control(itrace = ifelse(bDebug, 1, 0))
  
  ## Set random seed for reproducibility
  set.seed(nELSeed)
  
  ## Perform cross-validated glmnet model fitting
  cvfit <- cv.glmnet(
    train.x.net, 
    train.y.net, 
    offset = log(train.offset.net), 
    family = "poisson", 
    alpha = fGLMNetAlpha,
    parallel = TRUE
  )
  
  ## Stop the parallel cluster
  stopCluster(cl)
  
  ## Save the model to cache if caching is enabled
  if (bUseCache) {
    saveRDS(cvfit, paste0(cacheFileRoot, "_glmnet_int_cv_model.rds"))
  }
}

## Generate predictions for the test data (handling offsets correctly)
test[, predictions_glmnet := predict(
  cvfit,
  s = "lambda.min",
  newx = test.x.net, 
  type = "response", 
  newoffset = log(test.offset.net)
)]

## Generate predictions for the training data (handling offsets correctly)
train[, predictions_glmnet := predict(
  cvfit,
  s = "lambda.min",
  newx = train.x.net, 
  type = "response", 
  newoffset = log(train.offset.net)
)]

Model Plots and Tables

An important by-product of the n-fold cross validation is the plot of $\lambda$ values. The optimal choice of $\lambda$ is the lowest, and the model associated with that $\lambda$ is the final model.

One can also plot the trajectory of coefficients as progressively higher $\lambda$ imposes ever harsher penalties on the coefficients.

Lambda Plot

## plot lambda vs deviance

data.table(lambda=cvfit$lambda,
           cvm=cvfit$cvm,
           cvlo=cvfit$cvlo,
           cvup=cvfit$cvup,
           nzero=cvfit$nzero) %>%
  ggplot(aes(x=log(lambda))) +
  geom_point(aes(y=cvm),color="red") +
  geom_errorbar(aes(ymin=cvlo,ymax=cvup),color="grey") +
  scale_y_continuous(name=cvfit$name,
                     limits = range(cvfit$cvup,cvfit$cvlo)) +
  scale_x_continuous(name=expression(Log(lambda))) +
  geom_vline(xintercept=log(cvfit$lambda.min),linetype=3) +
  geom_vline(xintercept = log(cvfit$lambda.1se),linetype=3) +
  theme_minimal()

Coefficient Penalization

plot(cvfit$glmnet.fit,xvar="lambda")

Final Model

#---------------------------------------------------------------#
##### Examine model for minimum lambda and make predictions #####
#---------------------------------------------------------------#

## get coefficients for fitted version, reformatted and filtered
reformatCoefs(cvfit, pred_cols)  %>%
  filter(Coef != 0) %>%
  select(Feature1Name,
         Feature1Level,
         Feature2Name,
         Feature2Level,
         Coef) %>%
  mutate(Coef=exp(Coef)) %>%
  flextable() %>%
  highlight(j="Coef",color="yellow",part="body", i = ~ abs(log(Coef)) > log(1.05)) %>%
  set_formatter(
    Coef=function(x) paste0(sprintf("%.01f", 100*x),"%")
  ) %>%
  set_table_properties(opts_html=list(
        scroll=list(
          add_css="max-height: 500px;"
          )
        )
        ) %>%
  autofit() ->
  output_table

if(output_format == "html") {
  output_table
} else {
  wb <- wb_workbook()
  
  wb$add_worksheet("GLMNet Coefficients")
  
  wb <- wb_add_flextable(
    wb,
    "GLMNet Coefficients",
    output_table
  )
  
  wb$save(
    paste0(
      exportsRoot,
      "_glmnet_coefficients_table.xlsx"
    )
  )
  
  cat("See included Excel table for additional information.\n")
}

Feature1Name	Feature1Level	Feature2Name	Feature2Level	Coef
(Intercept)	(Intercept)			127.5%
uw	N/2/2			103.5%
uw	N/4/1			87.4%
uw	N/4/2			91.0%
face_amount_band	05 - 100,000 - 249,999			86.4%
face_amount_band	06 - 250,000 - 499,999			86.9%
face_amount_band	07 - 500,000 - 999,999			93.4%
face_amount_band	08 - 1,000,000+			96.3%
gender	M			106.9%
insurance_plan	xL			100.1%
uw	N/2/2	ia_band1	25-34	109.2%
uw	N/3/1	ia_band1	25-34	101.0%
uw	N/3/2	ia_band1	25-34	100.9%
uw	N/3/3	ia_band1	25-34	122.3%
uw	N/4/3	ia_band1	25-34	114.0%
uw	N/4/4	ia_band1	25-34	108.8%
uw	S/1/1	ia_band1	25-34	97.3%
uw	S/2/1	ia_band1	25-34	95.3%
uw	N/2/1	ia_band1	35-44	98.1%
uw	N/2/2	ia_band1	35-44	105.0%
uw	N/3/1	ia_band1	35-44	100.1%
uw	N/3/2	ia_band1	35-44	101.9%
uw	N/3/3	ia_band1	35-44	131.6%
uw	N/4/1	ia_band1	35-44	100.8%
uw	N/4/2	ia_band1	35-44	103.5%
uw	N/4/4	ia_band1	35-44	112.2%
uw	S/2/1	ia_band1	35-44	95.2%
uw	U/1/1	ia_band1	35-44	107.5%
uw	N/2/1	ia_band1	45-54	98.3%
uw	N/2/2	ia_band1	45-54	109.5%
uw	N/3/2	ia_band1	45-54	97.7%
uw	N/3/3	ia_band1	45-54	119.0%
uw	N/4/2	ia_band1	45-54	99.8%
uw	N/4/3	ia_band1	45-54	109.5%
uw	N/4/4	ia_band1	45-54	114.8%
uw	S/2/2	ia_band1	45-54	105.3%
uw	U/1/1	ia_band1	45-54	101.8%
uw	N/2/1	ia_band1	55-64	91.6%
uw	N/3/1	ia_band1	55-64	90.8%
uw	N/3/2	ia_band1	55-64	86.5%
uw	N/3/3	ia_band1	55-64	105.9%
uw	N/4/1	ia_band1	55-64	82.6%
uw	N/4/2	ia_band1	55-64	94.0%
uw	S/1/1	ia_band1	55-64	109.4%
uw	S/2/1	ia_band1	55-64	98.2%
uw	S/2/2	ia_band1	55-64	92.1%
uw	U/1/1	ia_band1	55-64	101.6%
uw	N/3/1	ia_band1	65-74	88.6%
uw	N/3/2	ia_band1	65-74	85.9%
uw	N/3/3	ia_band1	65-74	97.1%
uw	N/4/1	ia_band1	65-74	85.5%
uw	N/4/2	ia_band1	65-74	80.8%
uw	N/4/3	ia_band1	65-74	93.0%
uw	S/2/2	ia_band1	65-74	103.0%
uw	U/1/1	ia_band1	65-74	98.2%
uw	N/2/1	ia_band1	75-84	113.4%
uw	N/2/2	ia_band1	75-84	98.0%
uw	N/3/1	ia_band1	75-84	92.9%
uw	N/3/2	ia_band1	75-84	82.8%
uw	N/3/3	ia_band1	75-84	83.1%
uw	N/4/1	ia_band1	75-84	51.1%
uw	N/4/2	ia_band1	75-84	137.2%
uw	N/4/3	ia_band1	75-84	89.5%
uw	N/4/4	ia_band1	75-84	90.6%
uw	S/1/1	ia_band1	75-84	119.0%
uw	S/2/2	ia_band1	75-84	96.8%
uw	N/2/1	ia_band1	85-99	191.0%
uw	N/2/2	ia_band1	85-99	101.0%
uw	N/3/2	ia_band1	85-99	113.0%
uw	N/3/3	ia_band1	85-99	83.4%
uw	N/4/2	ia_band1	85-99	89.0%
uw	N/4/3	ia_band1	85-99	83.3%
uw	N/4/4	ia_band1	85-99	88.1%
uw	S/2/1	ia_band1	85-99	141.8%
uw	S/2/2	ia_band1	85-99	169.5%
uw	N/2/2	face_amount_band	04 - 50,000 - 99,999	100.8%
uw	N/3/3	face_amount_band	04 - 50,000 - 99,999	99.7%
uw	N/2/2	face_amount_band	05 - 100,000 - 249,999	101.0%
uw	N/4/1	face_amount_band	05 - 100,000 - 249,999	95.3%
uw	N/4/2	face_amount_band	05 - 100,000 - 249,999	96.2%
uw	S/2/1	face_amount_band	05 - 100,000 - 249,999	99.5%
uw	N/4/1	face_amount_band	06 - 250,000 - 499,999	98.9%
uw	N/4/3	face_amount_band	06 - 250,000 - 499,999	95.0%
uw	N/2/1	face_amount_band	07 - 500,000 - 999,999	99.3%
uw	N/2/2	face_amount_band	07 - 500,000 - 999,999	99.2%
uw	N/3/2	face_amount_band	07 - 500,000 - 999,999	99.8%
uw	N/3/3	face_amount_band	07 - 500,000 - 999,999	102.9%
uw	N/4/1	face_amount_band	07 - 500,000 - 999,999	98.4%
uw	N/4/3	face_amount_band	07 - 500,000 - 999,999	105.1%
uw	N/4/4	face_amount_band	07 - 500,000 - 999,999	106.6%
uw	S/1/1	face_amount_band	07 - 500,000 - 999,999	95.5%
uw	S/2/1	face_amount_band	07 - 500,000 - 999,999	100.9%
uw	S/2/2	face_amount_band	07 - 500,000 - 999,999	116.0%
uw	N/2/1	face_amount_band	08 - 1,000,000+	91.6%
uw	N/2/2	face_amount_band	08 - 1,000,000+	104.4%
uw	N/3/1	face_amount_band	08 - 1,000,000+	99.1%
uw	N/3/2	face_amount_band	08 - 1,000,000+	103.8%
uw	N/3/3	face_amount_band	08 - 1,000,000+	114.7%
uw	N/4/1	face_amount_band	08 - 1,000,000+	103.4%
uw	N/4/2	face_amount_band	08 - 1,000,000+	98.0%
uw	N/4/3	face_amount_band	08 - 1,000,000+	103.9%
uw	N/4/4	face_amount_band	08 - 1,000,000+	119.8%
uw	S/2/1	face_amount_band	08 - 1,000,000+	97.7%
uw	U/1/1	face_amount_band	08 - 1,000,000+	110.8%
face_amount_band	04 - 50,000 - 99,999	ia_band1	25-34	102.5%
face_amount_band	05 - 100,000 - 249,999	ia_band1	25-34	92.2%
face_amount_band	06 - 250,000 - 499,999	ia_band1	25-34	87.3%
face_amount_band	07 - 500,000 - 999,999	ia_band1	25-34	91.1%
face_amount_band	08 - 1,000,000+	ia_band1	25-34	87.3%
face_amount_band	04 - 50,000 - 99,999	ia_band1	35-44	107.4%
face_amount_band	05 - 100,000 - 249,999	ia_band1	35-44	97.5%
face_amount_band	06 - 250,000 - 499,999	ia_band1	35-44	93.3%
face_amount_band	07 - 500,000 - 999,999	ia_band1	35-44	92.0%
face_amount_band	08 - 1,000,000+	ia_band1	35-44	93.9%
face_amount_band	05 - 100,000 - 249,999	ia_band1	45-54	93.1%
face_amount_band	06 - 250,000 - 499,999	ia_band1	45-54	88.5%
face_amount_band	07 - 500,000 - 999,999	ia_band1	45-54	89.0%
face_amount_band	08 - 1,000,000+	ia_band1	45-54	90.1%
face_amount_band	04 - 50,000 - 99,999	ia_band1	55-64	98.1%
face_amount_band	05 - 100,000 - 249,999	ia_band1	65-74	106.4%
face_amount_band	06 - 250,000 - 499,999	ia_band1	65-74	109.3%
face_amount_band	07 - 500,000 - 999,999	ia_band1	65-74	112.6%
face_amount_band	08 - 1,000,000+	ia_band1	65-74	102.4%
face_amount_band	04 - 50,000 - 99,999	ia_band1	75-84	97.7%
face_amount_band	05 - 100,000 - 249,999	ia_band1	75-84	103.7%
face_amount_band	06 - 250,000 - 499,999	ia_band1	75-84	108.3%
face_amount_band	07 - 500,000 - 999,999	ia_band1	75-84	112.0%
face_amount_band	07 - 500,000 - 999,999	ia_band1	85-99	102.8%
face_amount_band	08 - 1,000,000+	ia_band1	85-99	85.5%
face_amount_band	08 - 1,000,000+	dur_band1	02	97.8%
face_amount_band	08 - 1,000,000+	dur_band1	03	99.0%
face_amount_band	05 - 100,000 - 249,999	dur_band1	04-05	98.4%
face_amount_band	06 - 250,000 - 499,999	dur_band1	04-05	98.5%
face_amount_band	05 - 100,000 - 249,999	dur_band1	06-15	96.8%
face_amount_band	07 - 500,000 - 999,999	dur_band1	06-15	97.2%
face_amount_band	08 - 1,000,000+	dur_band1	06-15	93.4%
face_amount_band	04 - 50,000 - 99,999	dur_band1	16-25	96.3%
face_amount_band	05 - 100,000 - 249,999	dur_band1	16-25	97.8%
face_amount_band	07 - 500,000 - 999,999	dur_band1	16-25	101.3%
face_amount_band	08 - 1,000,000+	dur_band1	16-25	106.4%
uw	N/2/1	dur_band1	02	143.5%
uw	N/3/2	dur_band1	02	102.1%
uw	N/4/1	dur_band1	02	81.4%
uw	N/4/3	dur_band1	02	113.4%
uw	N/2/2	dur_band1	03	117.3%
uw	N/3/1	dur_band1	03	107.0%
uw	N/4/1	dur_band1	03	98.9%
uw	N/4/2	dur_band1	03	84.4%
uw	N/4/4	dur_band1	03	102.6%
uw	N/2/1	dur_band1	04-05	81.0%
uw	N/3/1	dur_band1	04-05	96.9%
uw	N/3/3	dur_band1	04-05	106.8%
uw	N/4/3	dur_band1	04-05	99.0%
uw	S/2/1	dur_band1	04-05	97.7%
uw	N/2/1	dur_band1	06-15	96.5%
uw	N/2/2	dur_band1	06-15	100.6%
uw	N/3/1	dur_band1	06-15	105.4%
uw	N/3/2	dur_band1	06-15	103.3%
uw	N/4/3	dur_band1	06-15	107.4%
uw	N/4/4	dur_band1	06-15	101.4%
uw	S/2/1	dur_band1	06-15	88.8%
uw	U/1/1	dur_band1	06-15	102.5%
uw	N/2/1	dur_band1	16-25	100.5%
uw	N/2/2	dur_band1	16-25	101.7%
uw	N/3/1	dur_band1	16-25	97.1%
uw	N/3/2	dur_band1	16-25	90.9%
uw	N/3/3	dur_band1	16-25	88.1%
uw	N/4/2	dur_band1	16-25	97.2%
uw	N/4/4	dur_band1	16-25	81.6%
uw	S/1/1	dur_band1	16-25	108.2%
uw	S/2/2	dur_band1	16-25	103.8%
uw	U/1/1	dur_band1	16-25	101.3%
dur_band1	04-05	ia_band1	25-34	100.1%
dur_band1	16-25	ia_band1	25-34	99.3%
dur_band1	02	ia_band1	35-44	100.3%
dur_band1	03	ia_band1	35-44	101.9%
dur_band1	04-05	ia_band1	35-44	91.9%
dur_band1	06-15	ia_band1	35-44	93.6%
dur_band1	16-25	ia_band1	35-44	97.3%
dur_band1	02	ia_band1	45-54	97.8%
dur_band1	03	ia_band1	45-54	101.7%
dur_band1	16-25	ia_band1	45-54	102.9%
dur_band1	02	ia_band1	55-64	109.6%
dur_band1	03	ia_band1	55-64	92.9%
dur_band1	04-05	ia_band1	55-64	97.1%
dur_band1	06-15	ia_band1	55-64	100.0%
dur_band1	16-25	ia_band1	55-64	112.3%
dur_band1	04-05	ia_band1	65-74	101.5%
dur_band1	06-15	ia_band1	65-74	96.6%
dur_band1	16-25	ia_band1	65-74	103.2%
dur_band1	02	ia_band1	75-84	87.6%
dur_band1	04-05	ia_band1	75-84	113.2%
dur_band1	16-25	ia_band1	75-84	99.4%
dur_band1	04-05	ia_band1	85-99	82.9%
dur_band1	06-15	ia_band1	85-99	99.0%
dur_band1	16-25	ia_band1	85-99	56.1%
face_amount_band	04 - 50,000 - 99,999	gender	M	103.3%
face_amount_band	06 - 250,000 - 499,999	gender	M	93.3%
face_amount_band	07 - 500,000 - 999,999	gender	M	90.8%
face_amount_band	08 - 1,000,000+	gender	M	89.6%
dur_band1	04-05	ltp	10 yr	98.6%
dur_band1	06-15	ltp	10 yr	103.9%
dur_band1	16-25	ltp	10 yr	123.1%
dur_band1	03	ltp	15 yr	100.6%
dur_band1	04-05	ltp	15 yr	95.7%
dur_band1	06-15	ltp	15 yr	95.9%
dur_band1	16-25	ltp	15 yr	222.7%
dur_band1	06-15	ltp	20 yr	90.9%
dur_band1	16-25	ltp	20 yr	82.7%
dur_band1	04-05	ltp	25 yr	100.3%
dur_band1	06-15	ltp	25 yr	94.1%
dur_band1	16-25	ltp	25 yr	97.0%
dur_band1	04-05	ltp	30 yr	112.2%
dur_band1	16-25	ltp	30 yr	92.7%
dur_band1	04-05	ltp	Not Level Term	98.6%
face_amount_band	05 - 100,000 - 249,999	ltp	10 yr	99.1%
face_amount_band	06 - 250,000 - 499,999	ltp	10 yr	100.0%
face_amount_band	07 - 500,000 - 999,999	ltp	10 yr	103.0%
face_amount_band	08 - 1,000,000+	ltp	10 yr	98.0%
face_amount_band	04 - 50,000 - 99,999	ltp	15 yr	97.3%
face_amount_band	05 - 100,000 - 249,999	ltp	15 yr	92.0%
face_amount_band	08 - 1,000,000+	ltp	15 yr	102.2%
face_amount_band	05 - 100,000 - 249,999	ltp	20 yr	95.6%
face_amount_band	06 - 250,000 - 499,999	ltp	20 yr	98.7%
face_amount_band	07 - 500,000 - 999,999	ltp	20 yr	97.5%
face_amount_band	08 - 1,000,000+	ltp	20 yr	97.5%
face_amount_band	07 - 500,000 - 999,999	ltp	25 yr	96.6%
face_amount_band	05 - 100,000 - 249,999	ltp	30 yr	101.9%
face_amount_band	06 - 250,000 - 499,999	ltp	30 yr	99.6%
face_amount_band	07 - 500,000 - 999,999	ltp	30 yr	96.5%
face_amount_band	08 - 1,000,000+	ltp	30 yr	96.6%
face_amount_band	04 - 50,000 - 99,999	ltp	Not Level Term	90.2%
face_amount_band	05 - 100,000 - 249,999	ltp	Not Level Term	98.6%
face_amount_band	07 - 500,000 - 999,999	ltp	Not Level Term	98.3%
face_amount_band	04 - 50,000 - 99,999	ltp	Unknown	101.0%
face_amount_band	06 - 250,000 - 499,999	ltp	Unknown	95.7%
face_amount_band	07 - 500,000 - 999,999	ltp	Unknown	90.9%
uw	N/2/2	ltp	10 yr	109.2%
uw	N/3/3	ltp	10 yr	103.5%
uw	N/4/1	ltp	10 yr	108.2%
uw	N/4/3	ltp	10 yr	83.4%
uw	S/1/1	ltp	10 yr	108.0%
uw	S/2/2	ltp	10 yr	100.4%
uw	N/2/1	ltp	15 yr	98.1%
uw	N/3/1	ltp	15 yr	95.6%
uw	N/3/2	ltp	15 yr	89.6%
uw	N/4/1	ltp	15 yr	98.3%
uw	N/4/4	ltp	15 yr	104.5%
uw	N/2/2	ltp	20 yr	107.1%
uw	N/3/1	ltp	20 yr	91.9%
uw	N/3/2	ltp	20 yr	97.5%
uw	N/3/3	ltp	20 yr	106.0%
uw	N/4/2	ltp	20 yr	100.8%
uw	N/4/4	ltp	20 yr	108.5%
uw	S/1/1	ltp	20 yr	112.8%
uw	S/2/2	ltp	20 yr	105.5%
uw	N/2/2	ltp	25 yr	102.2%
uw	N/3/1	ltp	25 yr	91.9%
uw	N/3/2	ltp	25 yr	79.8%
uw	N/2/2	ltp	30 yr	99.8%
uw	N/3/1	ltp	30 yr	91.4%
uw	N/3/2	ltp	30 yr	83.9%
uw	N/3/3	ltp	30 yr	95.6%
uw	N/4/1	ltp	30 yr	95.5%
uw	N/4/3	ltp	30 yr	104.9%
uw	N/4/4	ltp	30 yr	103.0%
uw	N/2/1	ltp	Not Level Term	87.2%
uw	N/4/1	ltp	Not Level Term	99.9%
uw	N/4/2	ltp	Not Level Term	89.8%
uw	N/4/3	ltp	Not Level Term	81.3%
uw	S/2/1	ltp	Not Level Term	90.3%
uw	U/1/1	ltp	Not Level Term	112.4%
uw	N/2/1	ltp	Unknown	103.9%
uw	N/2/2	ltp	Unknown	116.5%
uw	N/3/1	ltp	Unknown	100.1%
uw	N/3/2	ltp	Unknown	113.9%
uw	N/3/3	ltp	Unknown	111.1%
uw	N/4/4	ltp	Unknown	86.8%
uw	S/1/1	ltp	Unknown	102.7%
face_amount_band	08 - 1,000,000+	iy_band1	1990-1999	93.6%
face_amount_band	05 - 100,000 - 249,999	iy_band1	2000-2009	98.5%
face_amount_band	06 - 250,000 - 499,999	iy_band1	2000-2009	93.7%
face_amount_band	07 - 500,000 - 999,999	iy_band1	2000-2009	89.1%
face_amount_band	08 - 1,000,000+	iy_band1	2000-2009	83.7%
face_amount_band	04 - 50,000 - 99,999	iy_band1	2010+	110.1%
face_amount_band	06 - 250,000 - 499,999	iy_band1	2010+	90.1%
face_amount_band	07 - 500,000 - 999,999	iy_band1	2010+	80.6%
face_amount_band	08 - 1,000,000+	iy_band1	2010+	71.6%
ia_band1	25-34	ltp	10 yr	100.2%
ia_band1	35-44	ltp	10 yr	100.9%
ia_band1	55-64	ltp	10 yr	93.1%
ia_band1	25-34	ltp	15 yr	99.4%
ia_band1	35-44	ltp	15 yr	104.0%
ia_band1	45-54	ltp	15 yr	93.5%
ia_band1	65-74	ltp	15 yr	117.0%
ia_band1	35-44	ltp	20 yr	99.2%
ia_band1	45-54	ltp	20 yr	95.3%
ia_band1	25-34	ltp	25 yr	98.4%
ia_band1	35-44	ltp	25 yr	97.5%
ia_band1	25-34	ltp	30 yr	96.8%
ia_band1	25-34	ltp	Not Level Term	100.1%
ia_band1	35-44	ltp	Not Level Term	104.2%
ia_band1	55-64	ltp	Not Level Term	96.1%
ia_band1	25-34	ltp	Unknown	89.8%
ia_band1	35-44	ltp	Unknown	94.1%
ia_band1	75-84	ltp	Unknown	158.4%
uw	N/2/2	insurance_plan	Perm	108.7%
uw	N/3/2	insurance_plan	Perm	102.1%
uw	N/3/3	insurance_plan	Perm	119.5%
uw	S/1/1	insurance_plan	Perm	95.4%
uw	U/1/1	insurance_plan	Perm	116.0%
uw	N/2/2	insurance_plan	Term	120.3%
uw	N/3/1	insurance_plan	Term	85.1%
uw	N/4/4	insurance_plan	Term	110.6%
uw	S/1/1	insurance_plan	Term	108.2%
uw	S/2/2	insurance_plan	Term	101.4%
uw	N/2/1	insurance_plan	xL	99.0%
uw	N/3/1	insurance_plan	xL	73.0%
uw	N/3/2	insurance_plan	xL	79.2%
uw	N/3/3	insurance_plan	xL	100.8%
uw	N/4/1	insurance_plan	xL	85.2%
uw	N/4/2	insurance_plan	xL	90.9%
uw	N/4/4	insurance_plan	xL	101.1%
uw	S/2/1	insurance_plan	xL	95.2%
uw	S/2/2	insurance_plan	xL	95.2%
ia_band1	25-34	gender	M	107.7%
ia_band1	35-44	gender	M	106.4%
ia_band1	45-54	gender	M	104.0%
ia_band1	65-74	gender	M	99.0%
ia_band1	75-84	gender	M	96.3%
ia_band1	85-99	gender	M	97.0%
ia_band1	35-44	iy_band1	1990-1999	96.0%
ia_band1	65-74	iy_band1	1990-1999	102.2%
ia_band1	75-84	iy_band1	1990-1999	99.5%
ia_band1	25-34	iy_band1	2000-2009	96.7%
ia_band1	55-64	iy_band1	2000-2009	96.8%
ia_band1	75-84	iy_band1	2000-2009	92.7%
ia_band1	85-99	iy_band1	2000-2009	97.1%
ia_band1	25-34	iy_band1	2010+	104.9%
ia_band1	35-44	iy_band1	2010+	107.2%
ia_band1	65-74	iy_band1	2010+	87.4%
uw	N/2/1	gender	M	104.1%
uw	N/2/2	gender	M	100.3%
uw	N/3/2	gender	M	103.2%
uw	N/3/3	gender	M	98.3%
uw	N/4/2	gender	M	111.9%
uw	N/4/3	gender	M	109.3%
uw	N/4/4	gender	M	101.2%
uw	S/1/1	gender	M	98.0%
uw	S/2/2	gender	M	98.5%
face_amount_band	04 - 50,000 - 99,999	insurance_plan	Perm	95.5%
face_amount_band	05 - 100,000 - 249,999	insurance_plan	Perm	93.3%
face_amount_band	06 - 250,000 - 499,999	insurance_plan	Perm	88.1%
face_amount_band	07 - 500,000 - 999,999	insurance_plan	Perm	80.5%
face_amount_band	04 - 50,000 - 99,999	insurance_plan	Term	102.0%
face_amount_band	06 - 250,000 - 499,999	insurance_plan	Term	96.0%
face_amount_band	07 - 500,000 - 999,999	insurance_plan	Term	91.7%
face_amount_band	08 - 1,000,000+	insurance_plan	Term	94.5%
ia_band1	25-34	insurance_plan	Perm	106.3%
ia_band1	45-54	insurance_plan	Perm	98.9%
ia_band1	55-64	insurance_plan	Perm	94.0%
ia_band1	75-84	insurance_plan	Perm	100.7%
ia_band1	85-99	insurance_plan	Perm	125.4%
ia_band1	25-34	insurance_plan	Term	99.6%
ia_band1	25-34	insurance_plan	xL	106.2%
ia_band1	35-44	insurance_plan	xL	102.0%
ia_band1	45-54	insurance_plan	xL	101.4%
ia_band1	65-74	insurance_plan	xL	94.4%
ia_band1	75-84	insurance_plan	xL	97.9%
ia_band1	85-99	insurance_plan	xL	90.4%
uw	N/2/2	iy_band1	1990-1999	107.1%
uw	N/3/2	iy_band1	1990-1999	112.0%
uw	N/4/1	iy_band1	1990-1999	94.2%
uw	N/4/2	iy_band1	1990-1999	94.5%
uw	S/2/1	iy_band1	1990-1999	112.5%
uw	S/2/2	iy_band1	1990-1999	118.0%
uw	N/2/1	iy_band1	2000-2009	91.7%
uw	N/2/2	iy_band1	2000-2009	103.2%
uw	N/3/1	iy_band1	2000-2009	97.8%
uw	N/3/3	iy_band1	2000-2009	105.2%
uw	N/4/1	iy_band1	2000-2009	98.8%
uw	S/1/1	iy_band1	2000-2009	101.6%
uw	S/2/2	iy_band1	2000-2009	109.9%
uw	N/2/1	iy_band1	2010+	87.9%
uw	N/3/3	iy_band1	2010+	100.9%
uw	S/2/1	iy_band1	2010+	95.8%
dur_band1	02	gender	M	102.6%
dur_band1	04-05	gender	M	103.8%
insurance_plan	Term	ltp	Not Level Term	75.8%
insurance_plan	xL	ltp	Not Level Term	100.4%
dur_band1	04-05	iy_band1	2000-2009	101.4%
dur_band1	06-15	iy_band1	2000-2009	99.7%
dur_band1	16-25	iy_band1	2000-2009	95.5%
dur_band1	04-05	iy_band1	2010+	97.8%
dur_band1	02	insurance_plan	Perm	129.7%
dur_band1	16-25	insurance_plan	Perm	87.3%
dur_band1	02	insurance_plan	xL	96.5%
dur_band1	03	insurance_plan	xL	109.2%
dur_band1	04-05	insurance_plan	xL	97.9%
dur_band1	06-15	insurance_plan	xL	104.0%
gender	M	insurance_plan	Perm	102.2%
ltp	20 yr	iy_band1	1990-1999	98.9%
ltp	25 yr	iy_band1	1990-1999	83.5%
ltp	30 yr	iy_band1	1990-1999	90.9%
ltp	Not Level Term	iy_band1	1990-1999	96.7%
ltp	Unknown	iy_band1	1990-1999	104.5%
ltp	15 yr	iy_band1	2000-2009	96.3%
ltp	Unknown	iy_band1	2000-2009	95.5%
insurance_plan	xL	iy_band1	1990-1999	96.8%
insurance_plan	xL	iy_band1	2000-2009	98.9%
insurance_plan	xL	iy_band1	2010+	103.7%
gender	M	iy_band1	1990-1999	96.3%
gender	M	iy_band1	2010+	103.5%
gender	M	ltp	15 yr	101.8%
gender	M	ltp	20 yr	101.0%

Lift

#-----------------------------------------#
##### Calculate Validation Metrics #####
#-----------------------------------------#

## generate plot
test[,decile.table(get(resp_var), predictions_glmnet/get(resp_offset), get(resp_offset))] %>%
  pivot_longer(-c(decile,exposures)) %>%
  as.data.table() %>%
  ggplot(aes(x=decile, y=value, col=name)) +  
  geom_line() +
  scale_x_continuous(breaks=c(1:10)) +
  labs(x="Decile",y="Deaths") +
  theme_minimal() +
  ggtitle("Decile Lift Plot")

Lorenz Curve

## lorenz plot
test[,lorenz(get(resp_var), predictions_glmnet / get(resp_offset), get(resp_offset))]

Plots of Terms

## Generate a grid of predictor levels for the training data
train[, ..pred_cols] %>%
  lapply(levels) %>%
  expand.grid() %>%
  setDT() -> train.grid

## Create model matrix for the training grid
train.grid %>%
  model.Matrix(
    object = glmnetFormula,
    data = .,
    sparse = bUseSparse
  ) %>%
  ## Predict new coefficients using the fitted cv.glmnet model
  predict(
    cvfit,
    newx = .,
    s = "lambda.min",
    newoffset = rep(0, nrow(.))
  ) %>%
  as.vector() -> newCoef

## Add the predicted factors to the training grid
train.grid %>%
  add_column(Factor = exp(newCoef)) %>%
  setDT() -> train.grid

## Reformat coefficients and generate the list of interactions
reformatCoefs(cvfit, pred_cols) %>%
  filter(Coef != 0 & !is.na(Feature2Name)) %>%
  select(Feature1Name, Feature2Name) %>%
  distinct() %>%
  as.list() %>%
  purrr::list_transpose() -> glmnet.int.list

Below are plots of the 2-way interaction terms, with external factors fixed at their middle values.

## Generate and print plots for each interaction pair in glmnet.int.list
glmnet.int.list %>%
  map(.f = \(x) {
    ## Generate a plot of cross-validated net coefficients
    plotCVNetCoefs(
      train.grid,
      sort(x),
      "Factor",
      pred_cols
    )
  }) %>%
  ## Set names for each plot based on the interaction pairs
  purrr::set_names(
    map(glmnet.int.list, \(x) paste0(x, collapse = " x "))
  ) %>%
  ## Print each plot with a corresponding title
  iwalk(~ {
    cat('#### ', .y, '\n\n')  # Print the plot title
    print(.x)  # Print the plot
    cat('\n\n')  # Add spacing after each plot
  })

uw x ia_band1

uw x face_amount_band

face_amount_band x ia_band1

face_amount_band x dur_band1

uw x dur_band1

dur_band1 x ia_band1

face_amount_band x gender

dur_band1 x ltp

face_amount_band x ltp

uw x ltp

face_amount_band x iy_band1

ia_band1 x ltp

uw x insurance_plan

ia_band1 x gender

ia_band1 x iy_band1

uw x gender

face_amount_band x insurance_plan

ia_band1 x insurance_plan

uw x iy_band1

dur_band1 x gender

insurance_plan x ltp

dur_band1 x iy_band1

dur_band1 x insurance_plan

gender x insurance_plan

ltp x iy_band1

insurance_plan x iy_band1

gender x iy_band1

gender x ltp

Tables of Terms

Below are tables of 2-way interactions, with external factors fixed at their middle values.

## Generate and format summary tables for each interaction pair in glmnet.int.list
glmnet.int.list %>%
  map(.f = \(x) {
    # Generate a table of cross-validated net coefficients
    tableCVNetCoefs(
      train.grid,
      sort(x),
      "Factor",
      pred.cols = pred_cols
    ) %>%
      ## Create a flextable from the generated table
      flextable() %>%
      ## Format the table values, converting numeric values to percentages
      set_formatter(values = function(x) {
        if (is.numeric(x))
          sprintf("%.1f%%", x * 100)
        else
          x
      }) %>%
      ## Set a caption for the table
      set_caption(
        paste0("Factors for ", x[1], " and ", x[2], ", all other factors fixed at middle levels")
      ) %>%
      ## Set table properties to enable scrolling
      set_table_properties(opts_html = list(
        scroll = list(
          add_css = "max-height: 500px;"
        )
      )) %>%
      autofit() #%>%
    ## Print the flextable
    #knitr::knit_print()
  }) %>%
  ## Set names for each table based on the interaction pairs
  purrr::set_names(
    map(glmnet.int.list, \(x) paste0(x, collapse = " x "))
  ) -> 
  output_tables

if(output_format == "html") {
  output_tables %>%
    map(.f=knitr::knit_print) %>%
    ## Generate a tabset from the list of tables
    generate_tabset(
      tabtitle = "",
      tablevel = 3
    ) %>%
    ## Print the generated tabset
    cat()
} else {
  export_tables_to_excel(
    output_tables,
    paste0(
      exportsRoot,
      "_glmnet_int_twoway_interaction_tables.xlsx"
    )
  )
  
  cat("See included Excel table for additional information.\n")
}

uw x ia_band1

Factors for uw and ia_band1, all other factors fixed at middle levels
ia_band1	N/1/1	N/2/1	N/2/2	N/3/1	N/3/2	N/3/3	N/4/1	N/4/2	N/4/3	N/4/4	S/1/1	S/2/1	S/2/2	U/1/1
18-24	97.1%	97.1%	148.3%	95.6%	108.3%	123.0%	75.3%	68.4%	97.1%	108.1%	104.5%	108.6%	120.9%	112.6%
25-34	95.2%	95.2%	158.7%	94.6%	107.0%	147.5%	73.9%	67.0%	108.6%	115.3%	99.6%	101.5%	118.5%	110.4%
35-44	91.9%	90.2%	147.3%	90.5%	104.4%	153.1%	71.9%	66.9%	91.9%	114.8%	98.8%	97.8%	114.3%	114.5%
45-54	86.7%	85.2%	144.9%	85.3%	94.4%	130.6%	67.2%	60.9%	94.9%	110.8%	93.2%	96.9%	113.6%	102.3%
55-64	84.8%	77.7%	129.5%	75.7%	81.8%	113.7%	54.3%	56.1%	84.8%	94.4%	99.8%	93.1%	97.2%	99.9%
65-74	105.5%	105.5%	161.2%	92.0%	101.1%	129.9%	70.0%	60.1%	98.2%	117.5%	113.6%	118.1%	135.3%	120.2%
75-84	100.9%	114.4%	150.9%	92.2%	93.1%	106.2%	40.0%	97.5%	90.3%	101.8%	129.2%	112.8%	121.5%	117.0%
85-99	121.8%	232.6%	187.8%	119.8%	153.4%	128.7%	94.5%	76.3%	101.5%	119.4%	131.0%	193.2%	256.9%	141.2%

uw x face_amount_band

Factors for uw and face_amount_band, all other factors fixed at middle levels
face_amount_band	N/1/1	N/2/1	N/2/2	N/3/1	N/3/2	N/3/3	N/4/1	N/4/2	N/4/3	N/4/4	S/1/1	S/2/1	S/2/2	U/1/1
01 - 0 - 49,999	120.8%	118.8%	200.1%	118.8%	131.6%	182.0%	98.4%	88.3%	132.2%	154.4%	129.9%	135.8%	158.3%	142.6%
04 - 50,000 - 99,999	115.3%	113.4%	192.5%	113.5%	125.6%	173.2%	93.9%	84.3%	126.2%	147.4%	124.1%	129.7%	151.1%	136.1%
05 - 100,000 - 249,999	86.7%	85.2%	144.9%	85.3%	94.4%	130.6%	67.2%	60.9%	94.9%	110.8%	93.2%	96.9%	113.6%	102.3%
06 - 250,000 - 499,999	80.8%	79.4%	133.8%	79.5%	88.0%	121.7%	65.1%	59.0%	84.0%	103.3%	86.9%	90.9%	105.9%	95.4%
07 - 500,000 - 999,999	78.8%	77.0%	129.5%	77.6%	85.7%	122.3%	63.2%	57.6%	90.7%	107.4%	81.1%	89.5%	119.9%	93.1%
08 - 1,000,000+	94.7%	85.3%	163.9%	92.4%	107.2%	163.7%	79.8%	67.8%	107.7%	145.1%	101.9%	104.1%	124.2%	123.9%

face_amount_band x ia_band1

Factors for face_amount_band and ia_band1, all other factors fixed at middle levels
face_amount_band	18-24	25-34	35-44	45-54	55-64	65-74	75-84	85-99
01 - 0 - 49,999	102.7%	109.2%	100.4%	98.4%	74.0%	89.7%	52.6%	128.7%
04 - 50,000 - 99,999	98.0%	106.8%	103.0%	93.9%	69.3%	85.7%	49.0%	122.9%
05 - 100,000 - 249,999	75.3%	73.9%	71.9%	67.2%	54.3%	70.0%	40.0%	94.5%
06 - 250,000 - 499,999	76.7%	71.3%	70.0%	65.1%	55.3%	73.3%	42.5%	96.2%
07 - 500,000 - 999,999	74.1%	71.7%	66.7%	63.2%	53.4%	72.9%	42.5%	95.5%
08 - 1,000,000+	92.4%	85.7%	84.8%	79.8%	66.6%	82.7%	47.3%	99.0%

face_amount_band x dur_band1

Factors for face_amount_band and dur_band1, all other factors fixed at middle levels
dur_band1	01 - 0 - 49,999	04 - 50,000 - 99,999	05 - 100,000 - 249,999	06 - 250,000 - 499,999	07 - 500,000 - 999,999	08 - 1,000,000+
01	97.9%	93.4%	66.9%	64.7%	62.9%	80.1%
02	101.0%	96.5%	69.1%	66.8%	64.9%	80.9%
03	98.4%	93.9%	67.2%	65.1%	63.2%	79.8%
04-05	97.9%	93.4%	65.8%	63.7%	62.9%	80.1%
06-15	88.9%	84.9%	58.9%	58.8%	55.6%	68.0%
16-25	72.7%	66.9%	48.6%	48.1%	47.3%	63.4%

uw x dur_band1

Factors for uw and dur_band1, all other factors fixed at middle levels
dur_band1	N/1/1	N/2/1	N/2/2	N/3/1	N/3/2	N/3/3	N/4/1	N/4/2	N/4/3	N/4/4	S/1/1	S/2/1	S/2/2	U/1/1
01	85.3%	83.8%	121.6%	78.4%	92.9%	128.4%	66.9%	71.0%	93.3%	106.2%	91.7%	95.4%	111.7%	100.6%
02	108.1%	152.6%	154.2%	99.4%	120.3%	162.9%	69.1%	90.0%	134.2%	134.7%	116.3%	120.9%	141.7%	127.6%
03	86.7%	85.2%	144.9%	85.3%	94.4%	130.6%	67.2%	60.9%	94.9%	110.8%	93.2%	96.9%	113.6%	102.3%
04-05	83.9%	66.8%	119.6%	74.7%	91.4%	135.0%	65.8%	69.8%	90.8%	104.5%	90.2%	91.7%	109.9%	99.0%
06-15	75.0%	71.2%	107.6%	72.7%	84.4%	113.0%	58.9%	62.5%	88.2%	94.8%	80.7%	74.5%	98.3%	90.8%
16-25	61.9%	61.2%	89.8%	55.3%	61.3%	82.2%	48.6%	50.1%	67.8%	62.9%	72.1%	69.3%	84.3%	74.1%

dur_band1 x ia_band1

Factors for dur_band1 and ia_band1, all other factors fixed at middle levels
dur_band1	18-24	25-34	35-44	45-54	55-64	65-74	75-84	85-99
01	76.2%	74.7%	71.3%	66.9%	59.1%	70.8%	40.4%	95.6%
02	80.5%	78.9%	75.6%	69.1%	68.4%	74.8%	37.4%	101.0%
03	75.3%	73.9%	71.9%	67.2%	54.3%	70.0%	40.0%	94.5%
04-05	75.0%	73.6%	64.5%	65.8%	56.5%	70.8%	45.0%	77.9%
06-15	67.1%	65.7%	58.8%	58.9%	52.0%	60.2%	35.6%	83.3%
16-25	53.8%	52.3%	49.0%	48.6%	46.8%	51.6%	28.4%	37.9%

face_amount_band x gender

Factors for face_amount_band and gender, all other factors fixed at middle levels
face_amount_band	F	M
01 - 0 - 49,999	98.4%	108.7%
04 - 50,000 - 99,999	93.9%	107.2%
05 - 100,000 - 249,999	67.2%	74.3%
06 - 250,000 - 499,999	65.1%	67.1%
07 - 500,000 - 999,999	63.2%	63.4%
08 - 1,000,000+	79.8%	79.0%

dur_band1 x ltp

Factors for dur_band1 and ltp, all other factors fixed at middle levels
dur_band1	5 yr	10 yr	15 yr	20 yr	25 yr	30 yr	Not Level Term	Unknown
01	74.3%	79.6%	62.8%	66.9%	62.0%	65.7%	70.8%	77.6%
02	76.7%	82.2%	64.8%	69.1%	64.0%	67.8%	73.1%	80.2%
03	74.6%	80.0%	63.5%	67.2%	62.3%	66.0%	71.1%	78.0%
04-05	73.1%	77.2%	59.1%	65.8%	61.2%	72.5%	68.6%	76.4%
06-15	71.9%	80.1%	58.3%	58.9%	56.5%	63.6%	68.5%	75.2%
16-25	65.2%	86.1%	122.9%	48.6%	52.8%	53.5%	62.2%	68.2%

face_amount_band x ltp

Factors for face_amount_band and ltp, all other factors fixed at middle levels
face_amount_band	5 yr	10 yr	15 yr	20 yr	25 yr	30 yr	Not Level Term	Unknown
01 - 0 - 49,999	104.4%	113.0%	96.6%	98.4%	87.2%	90.6%	100.9%	109.1%
04 - 50,000 - 99,999	99.7%	107.9%	89.7%	93.9%	83.2%	86.5%	86.9%	105.2%
05 - 100,000 - 249,999	74.6%	80.0%	63.5%	67.2%	62.3%	66.0%	71.1%	78.0%
06 - 250,000 - 499,999	70.0%	75.7%	64.7%	65.1%	58.4%	60.5%	67.6%	70.0%
07 - 500,000 - 999,999	68.8%	76.7%	63.6%	63.2%	55.5%	57.6%	65.4%	65.4%
08 - 1,000,000+	86.8%	92.1%	82.1%	79.8%	72.5%	72.8%	83.9%	90.8%

uw x ltp

Factors for uw and ltp, all other factors fixed at middle levels
ltp	N/1/1	N/2/1	N/2/2	N/3/1	N/3/2	N/3/3	N/4/1	N/4/2	N/4/3	N/4/4	S/1/1	S/2/1	S/2/2	U/1/1
5 yr	96.2%	94.6%	150.3%	103.0%	107.4%	136.8%	74.6%	67.1%	105.3%	113.4%	91.8%	107.6%	119.5%	113.6%
10 yr	95.3%	93.7%	162.5%	102.0%	106.4%	140.2%	80.0%	66.5%	87.0%	112.3%	98.2%	106.6%	118.9%	112.5%
15 yr	83.2%	80.3%	130.0%	85.2%	83.3%	118.4%	63.5%	58.1%	91.1%	102.5%	79.4%	93.1%	103.4%	98.3%
20 yr	86.7%	85.2%	144.9%	85.3%	94.4%	130.6%	67.2%	60.9%	94.9%	110.8%	93.2%	96.9%	113.6%	102.3%
25 yr	80.3%	79.0%	128.2%	79.0%	71.6%	114.2%	62.3%	56.0%	87.9%	94.7%	76.6%	89.9%	99.8%	94.8%
30 yr	89.1%	87.7%	138.9%	87.2%	83.5%	121.2%	66.0%	62.2%	102.4%	108.2%	85.1%	99.7%	110.7%	105.2%
Not Level Term	91.8%	78.7%	143.4%	98.2%	102.5%	130.5%	71.1%	57.5%	81.7%	108.2%	87.6%	92.7%	114.0%	121.8%
Unknown	100.6%	102.7%	183.0%	107.7%	127.9%	158.9%	78.0%	70.1%	110.1%	102.9%	98.5%	112.5%	124.9%	118.7%

face_amount_band x iy_band1

Factors for face_amount_band and iy_band1, all other factors fixed at middle levels
face_amount_band	1900-1989	1990-1999	2000-2009	2010+
01 - 0 - 49,999	105.6%	98.4%	104.3%	105.6%
04 - 50,000 - 99,999	100.8%	93.9%	99.6%	111.0%
05 - 100,000 - 249,999	72.2%	67.2%	70.3%	72.2%
06 - 250,000 - 499,999	69.8%	65.1%	64.7%	63.0%
07 - 500,000 - 999,999	67.8%	63.2%	59.7%	54.7%
08 - 1,000,000+	91.5%	79.8%	75.6%	65.5%

ia_band1 x ltp

Factors for ia_band1 and ltp, all other factors fixed at middle levels
ia_band1	5 yr	10 yr	15 yr	20 yr	25 yr	30 yr	Not Level Term	Unknown
18-24	79.7%	85.4%	72.5%	75.3%	66.5%	70.5%	75.9%	83.3%
25-34	78.1%	83.9%	70.7%	73.9%	64.2%	66.9%	74.5%	73.3%
35-44	76.7%	83.0%	72.6%	71.9%	62.4%	67.8%	76.1%	75.4%
45-54	74.6%	80.0%	63.5%	67.2%	62.3%	66.0%	71.1%	78.0%
55-64	57.5%	57.3%	52.3%	54.3%	48.0%	50.8%	52.6%	60.1%
65-74	74.1%	79.4%	78.9%	70.0%	61.9%	65.6%	70.6%	77.4%
75-84	42.3%	45.3%	38.5%	40.0%	35.3%	37.4%	40.3%	70.0%
85-99	100.0%	107.1%	90.9%	94.5%	83.5%	88.4%	95.3%	104.5%

uw x insurance_plan

Factors for uw and insurance_plan, all other factors fixed at middle levels
insurance_plan	N/1/1	N/2/1	N/2/2	N/3/1	N/3/2	N/3/3	N/4/1	N/4/2	N/4/3	N/4/4	S/1/1	S/2/1	S/2/2	U/1/1
Other	94.0%	92.4%	144.6%	92.5%	100.3%	118.4%	72.9%	66.1%	102.9%	120.1%	105.9%	105.1%	123.1%	95.6%
Perm	86.7%	85.2%	144.9%	85.3%	94.4%	130.6%	67.2%	60.9%	94.9%	110.8%	93.2%	96.9%	113.6%	102.3%
Term	94.0%	92.4%	173.9%	78.7%	100.3%	118.4%	72.9%	66.1%	102.9%	132.8%	114.6%	105.1%	124.9%	95.6%
xL	100.8%	98.2%	155.2%	72.4%	85.2%	128.1%	66.6%	64.5%	110.4%	130.3%	113.7%	107.4%	125.8%	102.7%

ia_band1 x gender

Factors for ia_band1 and gender, all other factors fixed at middle levels
gender	18-24	25-34	35-44	45-54	55-64	65-74	75-84	85-99
F	75.3%	73.9%	71.9%	67.2%	54.3%	70.0%	40.0%	94.5%
M	80.0%	84.5%	81.2%	74.3%	57.7%	73.7%	40.9%	97.4%

ia_band1 x iy_band1

Factors for ia_band1 and iy_band1, all other factors fixed at middle levels
ia_band1	1900-1989	1990-1999	2000-2009	2010+
18-24	80.9%	75.3%	78.7%	80.9%
25-34	79.3%	73.9%	74.6%	83.2%
35-44	80.3%	71.9%	78.2%	86.1%
45-54	72.2%	67.2%	70.3%	72.2%
55-64	58.3%	54.3%	55.0%	58.3%
65-74	73.6%	70.0%	71.7%	64.3%
75-84	43.1%	40.0%	38.9%	43.1%
85-99	101.4%	94.5%	95.9%	101.4%

uw x gender

Factors for uw and gender, all other factors fixed at middle levels
gender	N/1/1	N/2/1	N/2/2	N/3/1	N/3/2	N/3/3	N/4/1	N/4/2	N/4/3	N/4/4	S/1/1	S/2/1	S/2/2	U/1/1
F	86.7%	85.2%	144.9%	85.3%	94.4%	130.6%	67.2%	60.9%	94.9%	110.8%	93.2%	96.9%	113.6%	102.3%
M	95.7%	98.0%	160.6%	94.2%	107.6%	141.8%	74.3%	75.3%	114.6%	123.8%	100.9%	107.1%	123.6%	113.0%

face_amount_band x insurance_plan

Factors for face_amount_band and insurance_plan, all other factors fixed at middle levels
face_amount_band	Other	Perm	Term	xL
01 - 0 - 49,999	99.4%	98.4%	99.4%	90.9%
04 - 50,000 - 99,999	99.4%	93.9%	101.5%	90.9%
05 - 100,000 - 249,999	72.9%	67.2%	72.9%	66.6%
06 - 250,000 - 499,999	74.6%	65.1%	71.7%	68.2%
07 - 500,000 - 999,999	79.4%	63.2%	72.8%	72.5%
08 - 1,000,000+	80.6%	79.8%	76.2%	73.7%

ia_band1 x insurance_plan

Factors for ia_band1 and insurance_plan, all other factors fixed at middle levels
ia_band1	Other	Perm	Term	xL
18-24	80.8%	75.3%	80.8%	72.8%
25-34	74.5%	73.9%	74.1%	71.3%
35-44	77.1%	71.9%	77.1%	70.9%
45-54	72.9%	67.2%	72.9%	66.6%
55-64	62.0%	54.3%	62.0%	55.9%
65-74	75.1%	70.0%	75.1%	63.9%
75-84	42.6%	40.0%	42.6%	37.6%
85-99	80.8%	94.5%	80.8%	65.8%

uw x iy_band1

Factors for uw and iy_band1, all other factors fixed at middle levels
iy_band1	N/1/1	N/2/1	N/2/2	N/3/1	N/3/2	N/3/3	N/4/1	N/4/2	N/4/3	N/4/4	S/1/1	S/2/1	S/2/2	U/1/1
1900-1989	87.7%	86.2%	136.9%	86.3%	85.3%	132.1%	72.2%	65.2%	96.0%	112.1%	94.3%	87.2%	97.4%	103.5%
1990-1999	86.7%	85.2%	144.9%	85.3%	94.4%	130.6%	67.2%	60.9%	94.9%	110.8%	93.2%	96.9%	113.6%	102.3%
2000-2009	86.4%	77.9%	139.3%	83.2%	84.0%	136.9%	70.3%	64.3%	94.6%	110.4%	94.4%	85.9%	105.5%	102.0%
2010+	87.7%	75.8%	136.9%	86.3%	85.3%	133.2%	72.2%	65.2%	96.0%	112.1%	94.3%	83.5%	97.4%	103.5%

dur_band1 x gender

Factors for dur_band1 and gender, all other factors fixed at middle levels
dur_band1	F	M
01	66.9%	73.9%
02	69.1%	78.3%
03	67.2%	74.3%
04-05	65.8%	75.4%
06-15	58.9%	65.0%
16-25	48.6%	53.7%

insurance_plan x ltp

Factors for insurance_plan and ltp, all other factors fixed at middle levels
insurance_plan	5 yr	10 yr	15 yr	20 yr	25 yr	30 yr	Not Level Term	Unknown
Other	80.9%	86.7%	68.8%	72.9%	67.6%	71.6%	77.1%	84.6%
Perm	74.6%	80.0%	63.5%	67.2%	62.3%	66.0%	71.1%	78.0%
Term	80.9%	86.7%	68.8%	72.9%	67.6%	71.6%	58.5%	84.6%
xL	74.0%	79.3%	62.9%	66.6%	61.8%	65.4%	70.8%	77.3%

dur_band1 x iy_band1

Factors for dur_band1 and iy_band1, all other factors fixed at middle levels
dur_band1	1900-1989	1990-1999	2000-2009	2010+
01	71.8%	66.9%	69.9%	71.8%
02	74.1%	69.1%	72.2%	74.1%
03	72.2%	67.2%	70.3%	72.2%
04-05	70.6%	65.8%	69.8%	69.1%
06-15	63.2%	58.9%	61.4%	63.2%
16-25	52.2%	48.6%	48.5%	52.2%

dur_band1 x insurance_plan

Factors for dur_band1 and insurance_plan, all other factors fixed at middle levels
dur_band1	Other	Perm	Term	xL
01	72.5%	66.9%	72.5%	60.7%
02	57.7%	69.1%	57.7%	46.6%
03	72.9%	67.2%	72.9%	66.6%
04-05	71.3%	65.8%	71.3%	58.4%
06-15	63.8%	58.9%	63.8%	55.6%
16-25	60.4%	48.6%	60.4%	50.5%

gender x insurance_plan

Factors for gender and insurance_plan, all other factors fixed at middle levels
gender	Other	Perm	Term	xL
F	72.9%	67.2%	72.9%	66.6%
M	78.8%	74.3%	78.8%	72.0%

ltp x iy_band1

Factors for ltp and iy_band1, all other factors fixed at middle levels
iy_band1	5 yr	10 yr	15 yr	20 yr	25 yr	30 yr	Not Level Term	Unknown
1900-1989	79.2%	84.9%	67.4%	72.2%	79.2%	77.1%	78.0%	79.2%
1990-1999	74.6%	80.0%	63.5%	67.2%	62.3%	66.0%	71.1%	78.0%
2000-2009	77.1%	82.7%	63.2%	70.3%	77.1%	75.1%	76.0%	73.7%
2010+	79.2%	84.9%	67.4%	72.2%	79.2%	77.1%	78.0%	79.2%

insurance_plan x iy_band1

Factors for insurance_plan and iy_band1, all other factors fixed at middle levels
insurance_plan	1900-1989	1990-1999	2000-2009	2010+
Other	78.2%	72.9%	76.2%	78.2%
Perm	72.2%	67.2%	70.3%	72.2%
Term	78.2%	72.9%	76.2%	78.2%
xL	73.9%	66.6%	71.2%	76.6%

gender x iy_band1

Factors for gender and iy_band1, all other factors fixed at middle levels
gender	1900-1989	1990-1999	2000-2009	2010+
F	72.2%	67.2%	70.3%	72.2%
M	82.8%	74.3%	80.6%	85.7%

gender x ltp

Factors for gender and ltp, all other factors fixed at middle levels
gender	5 yr	10 yr	15 yr	20 yr	25 yr	30 yr	Not Level Term	Unknown
F	74.6%	80.0%	63.5%	67.2%	62.3%	66.0%	71.1%	78.0%
M	81.6%	87.5%	70.7%	74.3%	68.2%	72.2%	77.8%	85.3%

Goodness-of-Fit

Unvariate Goodness-of-Fit

## Generate and format summary tables for each factor column
map(factor_cols, .f = \(x) {
  ## Convert column name to symbol for tidy evaluation
  x <- sym(x)
  resp_var_sym <- resp_var
  
  ## Summarize data by the current factor column
  train %>%
    group_by(!!x) %>%
    summarize(
      Outcome = sum(amount_actual),  # Sum the actual amounts
      AM = sum(amount_actual) / sum(predictions_glmnet)  # Calculate the Actual-to-Model ratio
    ) %>%
    ## Create a flextable from the summarized data
    flextable() %>%
    ## Format the Actual-to-Model column as percentages
    set_formatter(
      AM = function(x) {
        if (is.numeric(x))
          sprintf("%.1f%%", x * 100)
        else
          x
      }
    ) %>%
    ## Format the Outcome column as numbers
    colformat_num(j = "Outcome") %>%
    ## Set header labels for the table
    set_header_labels(
      Outcome = "Outcome",
      AM = "Actual-to-Model"
    ) %>%
    autofit() #%>%
    # Print the flextable
    #knitr::knit_print()
}) %>%
  ## Set names for each table based on the factor columns
  purrr::set_names(factor_cols) ->
  output_tables

if(output_format == "html") {
  output_tables %>%
    map(.f=knitr::knit_print) %>%
    ## Generate a tabset from the list of tables
    generate_tabset(
      tabtitle = "",
      tablevel = 4
    ) %>%
    ## Print the generated tabset
    cat()
} else {
  export_tables_to_excel(
    output_tables,
    paste0(
      exportsRoot,
      "_glmnet_int_univariate_goodness_of_fit_tables.xlsx"
    )
  )
  
  cat("See included Excel table for additional information.\n")
}

uw

uw	Outcome	Actual-to-Model
N/1/1	19,203,624,009	100.0%
N/2/1	9,661,768,053	99.9%
N/2/2	11,546,531,146	100.2%
N/3/1	5,950,371,727	99.7%
N/3/2	8,810,968,555	99.9%
N/3/3	14,970,350,057	100.2%
N/4/1	6,842,959,121	99.6%
N/4/2	5,162,511,227	99.5%
N/4/3	3,565,224,892	99.9%
N/4/4	3,742,908,023	100.5%
S/1/1	4,365,025,076	100.5%
S/2/1	1,910,521,866	98.9%
S/2/2	1,704,933,729	100.8%
U/1/1	498,038,979	103.1%

face_amount_band

face_amount_band	Outcome	Actual-to-Model
01 - 0 - 49,999	4,193,809,072	103.7%
04 - 50,000 - 99,999	5,283,024,574	100.1%
05 - 100,000 - 249,999	16,241,007,604	99.7%
06 - 250,000 - 499,999	14,745,604,133	99.7%
07 - 500,000 - 999,999	15,180,744,107	99.8%
08 - 1,000,000+	42,291,546,970	99.9%

dur_band1

dur_band1	Outcome	Actual-to-Model
01	1,269,136,295	101.7%
02	1,852,767,081	99.2%
03	2,494,157,892	100.8%
04-05	6,321,445,332	99.8%
06-15	55,677,490,288	100.0%
16-25	30,320,739,572	100.0%

ia_band1

ia_band1	Outcome	Actual-to-Model
18-24	667,931,081	102.8%
25-34	6,267,527,815	99.9%
35-44	16,990,453,137	100.1%
45-54	20,629,510,315	100.0%
55-64	19,447,152,090	99.9%
65-74	16,608,837,516	100.0%
75-84	15,884,011,950	100.0%
85-99	1,440,312,556	99.7%

gender

gender	Outcome	Actual-to-Model
F	33,257,191,935	99.8%
M	64,678,544,525	100.1%

insurance_plan

insurance_plan	Outcome	Actual-to-Model
Other	187,352,300	100.5%
Perm	12,161,923,261	100.0%
Term	39,023,616,053	99.9%
xL	46,562,844,846	100.1%

ltp

ltp	Outcome	Actual-to-Model
5 yr	686,343,514	108.0%
10 yr	7,271,664,723	100.1%
15 yr	6,017,756,560	99.8%
20 yr	17,767,048,607	99.9%
25 yr	525,222,922	98.4%
30 yr	3,092,731,785	99.5%
Not Level Term	59,614,315,535	100.0%
Unknown	2,960,652,814	99.4%

iy_band1

iy_band1	Outcome	Actual-to-Model
1900-1989	1,464,179,514	103.2%
1990-1999	29,544,147,346	100.0%
2000-2009	55,045,293,308	99.9%
2010+	11,882,116,292	99.9%

Bivariate Goodness-of-Fit

## Create a list of unique pairs of factor columns
pairlist <- data.table()
for (i in 1:(length(factor_cols) - 1)) {
  for (j in (i + 1):length(factor_cols)) {
    if (i == 1 & j == 2) {
      pairlist <- data.table(F1 = factor_cols[i], F2 = factor_cols[j])
    } else {
      pairlist <- rbind(pairlist, data.table(F1 = factor_cols[i], F2 = factor_cols[j]))
    }
  }
}

## Generate and format summary tables for each pair of factor columns
map2(.x = pairlist$F1, .y = pairlist$F2, .f = \(x, y) {
  xs <- sym(x)
  ys <- sym(y)
  
  ## Choose grouping order based on the number of levels in each factor
  if (length(train[, levels(get(x))]) >= length(train[, levels(get(y))])) {
    fttmp <- train %>%
      group_by(!!xs, !!ys) %>%
      summarize(
        Outcome = sum(amount_actual),
        Ratio = sprintf("%.1f%%", 100 * sum(amount_actual) / sum(predictions_glmnet))
      ) %>%
      pivot_wider(
        names_from = !!ys,
        values_from = c(Outcome, Ratio),
        names_glue = paste0(y, ": {", y, "}.{.value}"),
        names_vary = "slowest"
      )
  } else {
    fttmp <- train %>%
      group_by(!!ys, !!xs) %>%
      summarize(
        Outcome = sum(amount_actual),
        Ratio = sprintf("%.1f%%", 100 * sum(amount_actual) / sum(predictions_glmnet))
      ) %>%
      pivot_wider(
        names_from = !!xs,
        values_from = c(Outcome, Ratio),
        names_glue = paste0(x, ": {", x, "}.{.value}"),
        names_vary = "slowest"
      )
  }
  
  ## Adjust column keys for the flextable
  fttmp.colkeys <- names(fttmp)[1]
  for (i in 1:((length(names(fttmp)) - 1) / 2)) {
    fttmp.colkeys <- c(fttmp.colkeys, paste0("blank", i), names(fttmp)[(2 * i):(2 * i + 1)])
  }
  
  ## Create and print the flextable
  fttmp %>%
    flextable(col_keys = fttmp.colkeys) %>%
    ftExtra::span_header(sep = "\\.") %>%
    align(align = 'center', part = "all") %>%
    empty_blanks() %>%
    autofit() #%>%
    #knitr::knit_print()
}) %>%
  ## Set names for each element in the list based on the factor column pairs
  purrr::set_names(pairlist[, paste0(F1, " x ", F2)]) ->
  output_tables

if(output_format == "html") {
  output_tables %>%
    map(.f=knitr::knit_print) %>%
    ## Generate a tabset from the list of formatted tables
    generate_tabset(
      tabtitle = "",
      tablevel = 4
    ) %>%
    ## Print the generated tabset
    cat()
} else {
  export_tables_to_excel(
    output_tables,
    paste0(
      exportsRoot,
      "_glmnet_int_bivariate_goodness_of_fit_tables.xlsx"
    )
  )
  
  cat("See included Excel table for additional information.\n")
}

uw x face_amount_band

uw	face_amount_band: 01 - 0 - 49,999		face_amount_band: 04 - 50,000 - 99,999		face_amount_band: 05 - 100,000 - 249,999		face_amount_band: 06 - 250,000 - 499,999		face_amount_band: 07 - 500,000 - 999,999		face_amount_band: 08 - 1,000,000+
uw	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
N/1/1	2,331,609,989	103.4%	2,665,573,911	101.2%	4,377,227,666	100.0%	2,484,222,240	99.5%	2,280,087,941	98.7%	5,064,902,262	98.6%
N/2/1	67,510,912	106.5%	227,491,364	103.1%	1,877,102,924	100.1%	1,581,998,866	99.8%	1,613,477,143	99.4%	4,294,186,844	99.8%
N/2/2	478,620,100	101.7%	716,716,216	101.7%	2,460,749,546	100.5%	1,807,687,840	99.7%	1,536,284,872	99.3%	4,546,472,572	100.2%
N/3/1	4,589,248	114.2%	36,548,841	96.9%	636,379,622	99.2%	1,099,392,788	99.6%	1,269,934,939	100.3%	2,903,526,289	99.6%
N/3/2	15,879,147	116.3%	84,477,290	110.9%	824,647,078	98.8%	1,194,151,445	99.2%	1,294,166,133	99.1%	5,397,647,462	100.2%
N/3/3	89,078,013	105.3%	266,651,094	95.9%	1,685,052,988	100.7%	1,897,715,243	99.8%	1,940,524,074	100.6%	9,091,328,645	100.1%
N/4/1	2,890,726	94.8%	23,168,994	88.2%	533,862,585	97.9%	1,152,655,538	99.0%	1,611,064,623	99.3%	3,519,316,655	100.3%
N/4/2	5,630,152	94.4%	43,592,461	88.3%	580,655,929	98.1%	959,152,506	101.2%	1,142,130,198	99.4%	2,431,349,981	99.6%
N/4/3	2,758,598	107.8%	19,493,024	84.7%	410,982,168	97.6%	652,807,204	98.4%	822,151,439	101.3%	1,657,032,459	100.6%
N/4/4	16,764,842	101.0%	69,558,070	96.3%	501,578,433	99.5%	668,382,546	100.3%	698,077,315	101.6%	1,788,546,817	100.6%
S/1/1	846,343,572	104.5%	864,666,577	99.7%	1,161,486,983	99.0%	510,128,489	100.9%	361,498,469	97.3%	620,900,986	100.9%
S/2/1	21,486,405	96.6%	85,321,233	92.5%	614,066,701	98.1%	400,530,746	99.7%	315,167,594	103.6%	473,949,187	97.9%
S/2/2	74,338,988	106.8%	135,934,467	92.6%	492,786,516	99.5%	294,838,420	102.0%	267,040,410	104.2%	439,994,928	101.2%
U/1/1	236,308,380	106.5%	43,831,032	90.9%	84,428,465	105.5%	41,940,262	95.9%	29,138,957	88.2%	62,391,883	111.2%

uw x dur_band1

uw	dur_band1: 01		dur_band1: 02		dur_band1: 03		dur_band1: 04-05		dur_band1: 06-15		dur_band1: 16-25
uw	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
N/1/1	107,404,206	98.1%	127,611,866	99.5%	186,488,501	107.0%	495,415,170	112.5%	4,962,783,774	98.8%	13,323,920,492	99.9%
N/2/1	38,939,578	73.6%	117,924,754	107.3%	103,706,358	106.5%	187,685,672	94.8%	4,423,311,154	99.7%	4,790,200,537	100.3%
N/2/2	68,820,764	97.5%	89,429,859	92.9%	153,874,204	106.5%	321,513,019	99.2%	6,248,105,591	100.2%	4,664,787,709	100.3%
N/3/1	91,917,216	86.7%	162,393,895	97.8%	239,046,418	104.2%	463,263,819	97.5%	4,110,289,163	100.3%	883,461,216	98.7%
N/3/2	73,068,462	89.0%	142,447,178	107.1%	172,423,532	98.0%	471,435,974	98.0%	7,112,246,310	100.2%	839,347,099	98.6%
N/3/3	209,144,931	114.5%	274,764,174	97.7%	374,385,325	101.0%	1,050,466,173	101.2%	12,379,149,521	100.0%	682,439,933	98.2%
N/4/1	174,316,019	95.1%	228,792,240	96.0%	389,498,840	97.6%	1,034,633,390	100.6%	4,624,074,611	99.8%	391,644,021	101.3%
N/4/2	115,650,082	106.5%	173,110,541	100.2%	196,392,926	95.4%	696,533,900	99.6%	3,717,590,891	99.7%	263,232,887	96.8%
N/4/3	97,264,439	90.3%	192,504,643	105.0%	206,119,588	98.7%	501,287,156	97.8%	2,450,944,955	100.5%	117,104,111	99.8%
N/4/4	153,464,379	115.1%	195,703,696	97.5%	283,447,385	103.6%	660,958,667	98.5%	2,334,500,075	100.6%	114,833,821	92.9%
S/1/1	30,145,775	135.7%	28,847,772	94.8%	40,393,654	105.9%	91,257,938	95.5%	1,121,084,058	100.4%	3,053,295,879	100.4%
S/2/1	42,753,182	109.0%	58,621,979	94.9%	74,856,996	93.5%	177,694,957	94.1%	1,089,203,131	98.8%	467,391,621	102.0%
S/2/2	55,705,006	139.4%	55,852,780	94.4%	68,480,349	97.6%	160,693,564	95.8%	985,575,007	99.9%	378,627,023	103.1%
U/1/1	10,542,256	105.7%	4,761,704	71.4%	5,043,816	83.1%	8,605,933	73.9%	118,632,047	109.6%	350,453,223	102.9%

uw x ia_band1

uw	ia_band1: 18-24		ia_band1: 25-34		ia_band1: 35-44		ia_band1: 45-54		ia_band1: 55-64		ia_band1: 65-74		ia_band1: 75-84		ia_band1: 85-99
uw	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
N/1/1	195,107,686	105.7%	1,468,766,511	96.5%	3,219,766,527	98.7%	3,573,783,086	98.8%	4,457,649,299	100.7%	4,148,815,415	101.4%	1,980,748,076	101.9%	158,987,409	98.6%
N/2/1	87,770,402	106.8%	804,579,965	100.9%	1,676,447,802	99.4%	1,660,839,957	99.4%	1,585,401,151	99.4%	2,098,151,266	100.0%	1,649,244,265	100.3%	99,333,245	102.4%
N/2/2	82,359,855	103.8%	734,969,884	101.6%	1,537,827,224	100.8%	1,935,817,830	100.6%	1,907,603,917	99.8%	2,320,063,956	99.7%	2,742,077,451	99.8%	285,811,029	101.4%
N/3/1	42,971,227	81.7%	542,310,521	102.1%	1,509,846,536	100.8%	1,517,047,115	99.7%	1,083,241,294	99.0%	682,592,797	98.6%	557,962,237	99.0%	14,400,000	101.3%
N/3/2	26,380,775	113.4%	339,627,773	103.4%	1,177,249,350	101.0%	1,552,812,212	99.3%	1,501,263,082	99.3%	1,589,866,999	99.4%	2,421,887,776	99.7%	201,880,588	101.3%
N/3/3	29,150,024	97.3%	391,634,706	103.1%	1,364,327,286	100.9%	1,932,181,705	100.6%	2,306,037,454	100.5%	2,995,484,451	99.7%	5,412,062,505	99.9%	539,471,926	99.2%
N/4/1	26,287,124	72.9%	602,385,596	100.0%	2,097,046,161	100.5%	2,253,038,301	99.9%	1,386,295,039	99.3%	431,665,885	98.0%	43,983,146	88.9%	2,257,869	114.2%
N/4/2	9,905,787	98.3%	233,732,519	96.5%	1,112,629,077	101.0%	1,595,224,194	99.3%	1,413,399,308	99.3%	448,165,240	98.0%	313,744,476	101.9%	35,710,626	92.6%
N/4/3	7,030,001	83.3%	209,711,793	105.0%	705,969,366	99.8%	1,147,207,102	100.9%	986,258,205	99.5%	356,988,649	97.7%	140,290,698	96.4%	11,769,078	82.7%
N/4/4	9,949,832	106.2%	167,099,067	106.8%	638,256,939	101.7%	1,003,674,951	101.1%	958,007,317	100.1%	551,614,283	99.1%	345,232,117	98.1%	69,073,517	95.1%
S/1/1	68,766,581	113.4%	451,737,170	97.4%	1,077,472,651	100.5%	1,226,674,054	100.6%	936,351,195	101.1%	461,642,247	99.8%	139,279,941	103.9%	3,101,237	72.1%
S/2/1	26,876,275	136.8%	179,096,556	94.0%	474,899,912	97.6%	608,218,812	100.7%	402,893,657	97.5%	174,002,125	101.9%	42,414,641	92.6%	2,119,888	188.8%
S/2/2	17,471,313	135.4%	117,414,041	102.5%	333,323,589	99.9%	526,461,474	102.3%	400,538,301	97.4%	233,737,976	103.7%	63,750,567	92.8%	12,236,468	118.0%
U/1/1	37,904,199	93.1%	24,461,713	113.1%	65,390,717	115.7%	96,529,522	109.8%	122,212,871	106.6%	116,046,227	94.9%	31,334,054	89.9%	4,159,676	89.9%

uw x gender

uw	gender: F		gender: M
uw	Outcome	Ratio	Outcome	Ratio
N/1/1	6,871,313,340	99.6%	12,332,310,669	100.2%
N/2/1	3,739,559,716	99.3%	5,922,208,337	100.3%
N/2/2	4,668,956,646	100.1%	6,877,574,500	100.3%
N/3/1	1,982,219,430	99.8%	3,968,152,297	99.6%
N/3/2	3,036,089,596	99.0%	5,774,878,959	100.3%
N/3/3	5,764,067,198	100.8%	9,206,282,859	99.8%
N/4/1	1,822,625,798	99.6%	5,020,333,323	99.7%
N/4/2	997,342,369	95.9%	4,165,168,858	100.4%
N/4/3	561,153,990	96.5%	3,004,070,902	100.6%
N/4/4	894,420,215	100.1%	2,848,487,808	100.7%
S/1/1	1,618,467,456	102.6%	2,746,557,620	99.3%
S/2/1	534,567,286	96.0%	1,375,954,580	100.1%
S/2/2	553,360,607	106.3%	1,151,573,122	98.4%
U/1/1	213,048,288	102.6%	284,990,691	103.4%

uw x insurance_plan

uw	insurance_plan: Other		insurance_plan: Perm		insurance_plan: Term		insurance_plan: xL
uw	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
N/1/1	41,338,662	125.0%	6,000,364,392	99.7%	3,545,626,085	97.6%	9,616,294,870	100.9%
N/2/1	10,081,665	102.2%	1,335,375,364	99.7%	2,232,053,273	100.2%	6,084,257,751	99.8%
N/2/2	15,784,451	88.8%	1,659,464,730	100.7%	3,001,480,499	100.8%	6,869,801,466	99.9%
N/3/1	40,123,942	233.9%	317,915,492	97.9%	3,500,287,255	99.4%	2,092,045,038	99.5%
N/3/2	20,508,375	93.9%	282,009,643	103.7%	3,266,885,153	99.8%	5,241,565,384	99.8%
N/3/3	32,085,711	112.9%	496,907,377	102.2%	4,052,652,325	100.0%	10,388,704,644	100.1%
N/4/1	2,300,000	30.0%	9,330,053	119.2%	6,251,798,169	99.8%	579,530,899	98.3%
N/4/2	7,223,579	69.2%	7,940,683	93.4%	4,382,666,145	99.8%	764,680,820	98.6%
N/4/3	103,684	2.1%	2,495,706	67.4%	3,036,110,841	100.3%	526,514,661	98.6%
N/4/4	3,739,130	21.5%	12,550,388	100.6%	2,742,467,039	100.8%	984,151,466	101.1%
S/1/1	3,801,542	139.7%	1,397,404,613	99.3%	1,062,735,519	102.2%	1,901,083,402	100.4%
S/2/1	6,544,922	114.8%	163,842,775	95.9%	1,131,185,266	99.6%	608,948,903	98.3%
S/2/2	3,059,454	35.1%	189,503,330	104.5%	778,511,625	102.8%	733,859,320	98.6%
U/1/1	657,183	79.6%	286,818,715	103.5%	39,156,859	100.6%	171,406,222	103.1%

uw x ltp

uw	ltp: 5 yr		ltp: 10 yr		ltp: 15 yr		ltp: 20 yr		ltp: 25 yr		ltp: 30 yr		ltp: Not Level Term		ltp: Unknown
uw	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
N/1/1	167,693,145	123.9%	329,259,367	90.7%	313,430,210	106.1%	1,254,368,612	96.0%	22,798,990	125.3%	140,376,344	99.8%	15,709,773,948	100.6%	1,265,923,393	95.3%
N/2/1	109,944,549	110.2%	251,674,560	102.2%	154,929,926	95.4%	1,003,102,210	99.3%	54,969,322	96.1%	91,526,385	98.3%	7,455,632,156	99.8%	539,988,945	101.7%
N/2/2	101,292,185	102.6%	464,466,296	102.1%	319,758,321	101.1%	1,222,999,988	100.8%	275,278,999	102.8%	237,099,784	96.9%	8,573,983,079	99.9%	351,652,494	102.9%
N/3/1	41,020,000	115.3%	402,661,935	98.5%	437,170,279	98.0%	1,876,102,407	99.5%	32,892,500	82.8%	371,739,248	98.0%	2,700,444,573	100.3%	88,340,785	109.9%
N/3/2	62,023,000	151.0%	523,988,195	99.8%	476,579,573	98.2%	1,824,845,270	99.5%	12,110,000	64.4%	210,771,725	96.5%	5,612,039,277	99.9%	88,611,515	110.2%
N/3/3	59,980,899	84.1%	896,107,443	101.1%	652,611,790	100.7%	1,842,591,351	100.5%	16,475,000	87.4%	201,629,249	96.0%	11,161,374,076	100.1%	139,580,249	106.7%
N/4/1	4,410,000	209.3%	1,142,209,199	100.8%	1,124,192,978	99.2%	3,080,000,938	99.8%	36,372,000	111.3%	845,189,202	99.1%	591,162,804	97.7%	19,422,000	83.0%
N/4/2	985,000	49.1%	899,393,774	99.5%	918,062,758	99.3%	2,118,167,322	100.4%	14,432,100	86.9%	414,673,900	100.3%	779,845,082	98.2%	16,951,291	72.8%
N/4/3	300,000	41.4%	711,850,279	98.7%	663,787,225	100.8%	1,348,193,434	99.9%	8,775,000	133.1%	270,343,655	103.0%	529,114,051	97.6%	32,861,248	120.2%
N/4/4	1,900,000	83.0%	861,440,026	100.5%	549,878,289	101.8%	1,117,119,304	100.8%	8,832,928	86.8%	186,021,492	104.7%	1,000,440,984	99.7%	17,275,000	69.7%
S/1/1	77,649,707	101.0%	220,477,191	104.3%	92,140,018	92.4%	325,974,726	102.8%	37,348,584	94.1%	43,126,718	112.2%	3,324,217,468	100.0%	244,090,664	103.9%
S/2/1	35,606,411	82.4%	312,687,204	100.6%	190,844,063	99.6%	450,793,441	101.0%	2,925,000	97.0%	50,493,377	106.9%	784,528,034	98.0%	82,644,336	93.5%
S/2/2	19,448,975	86.9%	251,942,737	103.8%	122,991,130	100.2%	294,652,404	103.2%	1,050,000	58.1%	28,434,706	106.0%	932,515,030	99.0%	53,898,747	113.7%
U/1/1	4,089,643	105.2%	3,506,517	95.5%	1,380,000	26.5%	8,137,200	135.6%	962,499	33.6%	1,306,000	85.4%	459,244,973	103.3%	19,412,147	125.7%

uw x iy_band1

uw	iy_band1: 1900-1989		iy_band1: 1990-1999		iy_band1: 2000-2009		iy_band1: 2010+
uw	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
N/1/1	982,160,538	103.0%	12,534,732,161	99.7%	4,842,664,276	99.4%	844,067,034	104.5%
N/2/1	112,802,132	111.3%	4,655,974,027	100.2%	4,443,736,468	99.7%	449,255,426	96.4%
N/2/2	107,232,938	106.9%	4,512,424,274	100.3%	6,312,557,239	100.2%	614,316,695	98.8%
N/3/1	282,000	22.4%	1,044,203,844	99.2%	3,887,317,728	99.7%	1,018,568,155	100.5%
N/3/2	250,000	15.1%	1,008,204,301	101.2%	6,934,988,157	99.9%	867,526,097	98.0%
N/3/3	4,055,561	33.6%	755,727,777	99.6%	12,308,187,396	100.1%	1,902,379,323	101.1%
N/4/1	0	0.0%	394,664,512	97.8%	4,606,160,597	99.7%	1,842,134,012	99.8%
N/4/2			308,834,925	97.3%	3,702,484,884	99.7%	1,151,191,418	99.5%
N/4/3			158,854,990	103.3%	2,386,096,189	99.5%	1,020,273,713	100.3%
N/4/4	0	0.0%	146,243,472	98.4%	2,314,552,528	100.3%	1,282,112,023	101.2%
S/1/1	212,607,764	102.1%	2,836,569,692	100.1%	1,124,908,213	101.2%	190,939,407	100.5%
S/2/1	7,889,930	105.4%	479,056,419	102.4%	1,077,182,521	98.8%	346,392,996	94.8%
S/2/2	7,665,540	130.5%	388,469,845	102.9%	986,492,416	101.4%	322,305,928	96.2%
U/1/1	29,233,111	112.9%	320,187,107	102.1%	117,964,696	109.5%	30,654,065	85.2%

face_amount_band x dur_band1

face_amount_band	dur_band1: 01		dur_band1: 02		dur_band1: 03		dur_band1: 04-05		dur_band1: 06-15		dur_band1: 16-25
face_amount_band	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
01 - 0 - 49,999	37,859,834	193.5%	46,746,236	142.4%	59,349,580	141.6%	129,333,013	124.3%	856,334,359	108.6%	3,064,186,050	100.3%
04 - 50,000 - 99,999	52,511,061	130.3%	66,787,618	108.1%	90,269,149	111.6%	206,424,919	107.5%	1,469,057,939	98.8%	3,397,973,888	99.5%
05 - 100,000 - 249,999	191,101,070	106.9%	269,652,985	97.5%	371,980,209	102.4%	855,544,316	97.8%	7,074,965,104	99.7%	7,477,763,920	99.8%
06 - 250,000 - 499,999	209,469,031	100.9%	304,453,835	96.4%	416,690,820	100.8%	1,000,438,105	98.2%	8,109,440,540	99.7%	4,705,111,802	100.1%
07 - 500,000 - 999,999	215,479,316	92.7%	350,555,118	99.8%	461,397,659	99.1%	1,167,345,786	99.0%	8,791,983,362	99.8%	4,193,982,866	100.4%
08 - 1,000,000+	562,715,983	98.9%	814,571,289	98.3%	1,094,470,475	98.7%	2,962,359,193	100.0%	29,375,708,984	100.0%	7,481,721,046	100.2%

face_amount_band x ia_band1

ia_band1	face_amount_band: 01 - 0 - 49,999		face_amount_band: 04 - 50,000 - 99,999		face_amount_band: 05 - 100,000 - 249,999		face_amount_band: 06 - 250,000 - 499,999		face_amount_band: 07 - 500,000 - 999,999		face_amount_band: 08 - 1,000,000+
ia_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
18-24	75,552,784	127.6%	133,231,819	113.5%	220,299,419	97.6%	115,618,404	89.3%	61,754,428	97.5%	61,474,227	112.3%
25-34	192,799,300	136.6%	458,412,108	103.6%	1,476,938,558	98.7%	1,398,298,659	98.7%	1,378,321,244	98.7%	1,362,757,946	98.7%
35-44	421,318,794	123.4%	898,534,810	101.8%	3,004,123,357	99.4%	3,258,823,688	99.5%	3,627,163,035	99.5%	5,780,489,453	99.7%
45-54	861,129,196	108.8%	1,234,583,452	99.7%	4,027,597,384	99.5%	3,776,175,782	99.5%	3,750,396,726	99.5%	6,979,627,775	99.8%
55-64	1,371,789,823	99.8%	1,302,388,605	98.8%	4,041,501,912	99.8%	3,254,795,121	100.1%	2,876,205,825	99.7%	6,600,470,804	100.1%
65-74	1,035,580,594	95.6%	950,356,588	98.8%	2,489,528,859	100.6%	1,929,056,738	100.7%	2,107,668,820	100.6%	8,096,645,917	100.2%
75-84	221,626,122	94.5%	285,327,929	96.9%	901,837,477	101.1%	915,794,848	101.0%	1,257,117,333	100.6%	12,302,308,241	99.9%
85-99	14,012,459	80.9%	20,189,263	86.9%	79,180,638	100.0%	97,040,893	101.9%	122,116,696	103.3%	1,107,772,607	99.6%

face_amount_band x gender

face_amount_band	gender: F		gender: M
face_amount_band	Outcome	Ratio	Outcome	Ratio
01 - 0 - 49,999	2,082,160,405	101.0%	2,111,648,667	106.5%
04 - 50,000 - 99,999	2,075,208,647	99.0%	3,207,815,927	100.9%
05 - 100,000 - 249,999	5,592,847,725	99.5%	10,648,159,879	99.8%
06 - 250,000 - 499,999	4,715,664,348	99.8%	10,029,939,785	99.7%
07 - 500,000 - 999,999	4,435,784,376	99.8%	10,744,959,731	99.7%
08 - 1,000,000+	14,355,526,434	100.0%	27,936,020,536	99.9%

face_amount_band x insurance_plan

face_amount_band	insurance_plan: Other		insurance_plan: Perm		insurance_plan: Term		insurance_plan: xL
face_amount_band	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
01 - 0 - 49,999	4,987,841	101.9%	2,341,324,220	102.9%	430,107,626	113.7%	1,417,389,385	102.3%
04 - 50,000 - 99,999	5,666,783	104.3%	1,269,348,289	98.8%	1,013,673,047	103.1%	2,994,336,455	99.7%
05 - 100,000 - 249,999	34,041,514	113.1%	2,315,213,211	99.4%	6,874,173,321	99.6%	7,017,579,558	99.9%
06 - 250,000 - 499,999	28,639,038	132.3%	1,510,214,779	99.0%	8,348,184,153	99.6%	4,858,566,163	100.0%
07 - 500,000 - 999,999	18,132,026	94.5%	1,434,005,349	99.0%	8,575,363,506	99.6%	5,153,243,226	100.2%
08 - 1,000,000+	95,885,098	91.2%	3,291,817,413	99.8%	13,782,114,400	99.8%	25,121,730,059	100.0%

face_amount_band x ltp

ltp	face_amount_band: 01 - 0 - 49,999		face_amount_band: 04 - 50,000 - 99,999		face_amount_band: 05 - 100,000 - 249,999		face_amount_band: 06 - 250,000 - 499,999		face_amount_band: 07 - 500,000 - 999,999		face_amount_band: 08 - 1,000,000+
ltp	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
5 yr	40,106,609	113.4%	108,758,076	106.5%	223,852,354	106.4%	134,040,438	109.8%	88,356,037	115.8%	91,230,000	102.5%
10 yr	79,471,212	101.4%	152,307,148	101.6%	1,212,386,424	98.8%	1,312,587,882	101.1%	1,417,042,517	101.0%	3,097,869,540	99.6%
15 yr	63,154,393	107.0%	171,454,282	92.8%	1,115,915,003	98.7%	1,284,208,527	99.2%	1,254,669,304	100.8%	2,128,355,051	100.6%
20 yr	113,505,987	120.7%	278,179,336	104.4%	2,893,988,498	99.5%	3,894,785,171	99.6%	4,168,688,693	99.7%	6,417,900,922	99.8%
25 yr	9,580,391	164.9%	26,792,196	123.0%	154,206,000	101.4%	162,940,708	96.7%	101,887,627	90.5%	69,816,000	95.4%
30 yr	6,329,793	169.8%	31,267,129	124.0%	436,286,383	103.0%	721,667,111	98.3%	809,282,434	98.5%	1,087,898,935	99.0%
Not Level Term	3,802,855,700	102.8%	4,317,239,186	99.4%	9,477,900,932	99.7%	6,615,106,546	99.9%	6,758,480,601	99.6%	28,642,732,570	100.0%
Unknown	78,804,987	115.8%	197,027,221	106.3%	726,472,010	99.4%	620,267,750	97.8%	582,336,894	97.8%	755,743,952	99.0%

face_amount_band x iy_band1

face_amount_band	iy_band1: 1900-1989		iy_band1: 1990-1999		iy_band1: 2000-2009		iy_band1: 2010+
face_amount_band	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
01 - 0 - 49,999	267,683,109	95.6%	2,808,814,596	100.9%	833,122,239	108.0%	284,189,128	135.7%
04 - 50,000 - 99,999	267,216,839	98.0%	3,149,056,719	99.6%	1,444,089,389	99.6%	422,661,627	107.2%
05 - 100,000 - 249,999	404,345,538	100.0%	7,205,856,453	99.9%	6,896,526,088	99.7%	1,734,279,525	99.1%
06 - 250,000 - 499,999	158,023,865	101.7%	4,708,333,873	100.3%	7,919,827,744	99.7%	1,959,418,651	98.4%
07 - 500,000 - 999,999	136,967,257	119.4%	4,219,050,921	99.8%	8,611,625,509	99.8%	2,213,100,420	98.6%
08 - 1,000,000+	229,942,906	120.0%	7,453,034,784	99.8%	29,340,102,339	99.9%	5,268,466,941	99.4%

dur_band1 x ia_band1

ia_band1	dur_band1: 01		dur_band1: 02		dur_band1: 03		dur_band1: 04-05		dur_band1: 06-15		dur_band1: 16-25
ia_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
18-24	38,014,843	86.3%	30,257,763	72.7%	27,602,930	83.2%	48,778,652	91.5%	247,658,415	116.5%	275,618,478	104.0%
25-34	134,511,109	99.3%	163,517,330	100.5%	194,704,716	105.6%	424,147,771	104.2%	2,807,372,903	99.5%	2,543,273,986	99.3%
35-44	290,877,481	122.9%	372,345,417	104.0%	502,388,214	102.9%	1,120,149,604	98.4%	8,625,261,989	99.8%	6,079,430,432	99.7%
45-54	284,754,759	90.1%	443,414,694	96.9%	668,187,442	102.2%	1,648,875,363	99.6%	10,888,781,617	100.1%	6,695,496,440	100.3%
55-64	316,638,372	99.8%	537,383,357	102.7%	610,988,925	97.8%	1,580,728,594	98.9%	9,237,741,705	99.8%	7,163,671,137	100.2%
65-74	143,303,417	95.4%	240,017,175	96.0%	343,292,965	100.2%	930,432,092	101.6%	8,689,000,890	99.8%	6,262,790,977	100.2%
75-84	45,633,095	117.0%	49,322,782	85.7%	121,205,864	98.0%	506,866,160	102.0%	13,883,785,714	100.0%	1,277,198,335	99.5%
85-99	15,403,219	171.3%	16,508,563	101.1%	25,786,836	110.5%	61,467,096	91.9%	1,297,887,055	99.5%	23,259,787	90.2%

dur_band1 x gender

dur_band1	gender: F		gender: M
dur_band1	Outcome	Ratio	Outcome	Ratio
01	322,942,506	116.2%	946,193,789	97.5%
02	424,637,814	91.6%	1,428,129,267	101.7%
03	697,046,249	101.1%	1,797,111,643	100.7%
04-05	1,788,538,549	97.9%	4,532,906,783	100.6%
06-15	19,652,776,350	99.8%	36,024,713,938	100.1%
16-25	10,371,250,467	100.2%	19,949,489,105	99.9%

dur_band1 x insurance_plan

dur_band1	insurance_plan: Other		insurance_plan: Perm		insurance_plan: Term		insurance_plan: xL
dur_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
01	6,987,296	74.8%	115,387,649	123.3%	816,300,396	96.9%	330,460,954	109.3%
02	12,956,089	97.1%	186,759,670	107.8%	1,193,573,840	98.8%	459,477,482	97.1%
03	10,940,700	66.9%	173,260,241	105.6%	1,539,568,218	100.2%	770,388,733	101.8%
04-05	41,382,619	117.0%	351,202,533	96.0%	3,960,972,351	100.4%	1,967,887,829	99.1%
06-15	92,001,723	94.9%	2,251,458,889	99.4%	24,546,892,971	99.9%	28,787,136,705	100.1%
16-25	23,083,873	153.3%	9,083,854,279	99.8%	6,966,308,277	100.1%	14,247,493,143	100.1%

dur_band1 x ltp

ltp	dur_band1: 01		dur_band1: 02		dur_band1: 03		dur_band1: 04-05		dur_band1: 06-15		dur_band1: 16-25
ltp	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
5 yr	8,474,532	141.6%	5,768,265	73.7%	13,120,000	106.1%	47,280,625	109.3%	258,887,272	107.3%	352,812,820	108.7%
10 yr	250,039,805	98.0%	374,445,553	99.4%	463,240,176	98.8%	1,216,385,768	98.8%	4,556,704,077	100.4%	410,849,344	103.6%
15 yr	106,613,913	100.1%	154,795,986	95.4%	219,264,610	105.5%	520,402,972	97.4%	4,466,179,563	99.6%	550,499,516	102.9%
20 yr	314,198,709	99.9%	447,558,227	99.0%	600,176,032	100.9%	1,543,338,839	100.5%	11,599,090,681	99.9%	3,262,686,119	99.5%
25 yr	3,615,999	44.5%	13,707,800	129.4%	18,921,628	118.2%	49,784,528	127.3%	267,598,048	95.6%	171,594,919	95.2%
30 yr	66,053,409	84.1%	98,748,126	100.4%	130,459,928	96.2%	388,766,012	103.3%	1,983,373,230	100.1%	425,331,080	97.4%
Not Level Term	453,235,980	111.8%	659,812,994	100.0%	954,906,252	101.9%	2,361,669,130	98.9%	31,616,749,475	100.0%	23,567,941,704	100.0%
Unknown	66,903,948	90.9%	97,930,130	98.4%	94,069,266	92.6%	193,817,458	105.2%	928,907,942	99.3%	1,579,024,070	99.7%

dur_band1 x iy_band1

dur_band1	iy_band1: 2010+		iy_band1: 2000-2009		iy_band1: 1990-1999		iy_band1: 1900-1989
dur_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
01	1,269,136,295	101.7%
02	1,852,767,081	99.2%
03	2,494,157,892	100.8%
04-05	4,479,996,408	99.3%	1,841,448,924	101.3%
06-15	1,786,058,616	99.9%	51,370,835,971	99.9%	2,520,595,701	100.4%
16-25			1,833,008,413	98.8%	27,023,551,645	100.0%	1,464,179,514	103.2%

ia_band1 x gender

ia_band1	gender: F		gender: M
ia_band1	Outcome	Ratio	Outcome	Ratio
18-24	210,351,040	93.6%	457,580,041	107.6%
25-34	1,919,168,906	98.3%	4,348,358,909	100.6%
35-44	4,376,241,525	99.9%	12,614,211,612	100.2%
45-54	4,611,772,525	99.3%	16,017,737,790	100.2%
55-64	4,809,017,158	100.0%	14,638,134,932	99.8%
65-74	7,381,394,530	100.2%	9,227,442,986	99.7%
75-84	9,045,279,898	100.1%	6,838,732,052	99.8%
85-99	903,966,353	100.3%	536,346,203	98.7%

ia_band1 x insurance_plan

ia_band1	insurance_plan: Other		insurance_plan: Perm		insurance_plan: Term		insurance_plan: xL
ia_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
18-24	708,665	57.4%	166,230,095	97.5%	267,466,700	100.1%	233,525,621	110.7%
25-34	3,256,794	65.1%	852,327,846	101.7%	4,057,228,956	99.2%	1,354,714,219	101.1%
35-44	21,307,402	110.1%	1,846,365,837	100.1%	11,715,945,894	100.1%	3,406,834,004	100.4%
45-54	57,525,597	129.7%	2,565,363,967	99.5%	12,641,031,673	99.9%	5,365,589,078	100.3%
55-64	47,144,955	81.0%	3,306,501,193	99.6%	8,357,779,097	99.9%	7,735,726,845	100.1%
65-74	35,749,661	103.7%	2,847,699,847	100.0%	1,896,982,781	100.4%	11,828,405,227	99.9%
75-84	21,392,494	97.5%	492,087,214	102.3%	86,369,648	101.6%	15,284,162,594	99.9%
85-99	266,732	14.7%	85,347,262	107.3%	811,304	108.2%	1,353,887,258	99.3%

ia_band1 x ltp

ia_band1	ltp: 5 yr		ltp: 10 yr		ltp: 15 yr		ltp: 20 yr		ltp: 25 yr		ltp: 30 yr		ltp: Not Level Term		ltp: Unknown
ia_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
18-24	17,726,831	95.3%	32,009,383	100.3%	3,343,000	42.3%	80,676,936	111.9%	13,002,500	98.0%	48,824,425	104.2%	403,156,010	104.9%	69,191,996	92.5%
25-34	147,704,951	111.5%	317,080,031	104.5%	150,181,284	91.9%	1,670,129,784	99.7%	164,417,165	93.5%	748,387,532	98.2%	2,274,993,951	101.0%	794,633,117	98.3%
35-44	232,956,798	104.0%	1,049,894,842	101.4%	983,755,234	101.4%	6,112,699,729	99.8%	222,687,940	95.1%	1,620,869,320	100.2%	5,508,205,763	100.4%	1,259,383,511	99.0%
45-54	178,555,603	110.1%	2,312,649,990	100.2%	2,156,659,786	99.4%	6,421,617,312	99.8%	95,336,910	109.7%	652,477,911	99.1%	8,212,889,839	100.0%	599,322,964	100.0%
55-64	76,414,209	103.6%	2,489,338,251	99.5%	2,056,964,977	99.7%	3,341,836,281	100.2%	27,176,407	125.5%	22,072,597	100.9%	11,252,197,601	99.8%	181,151,767	104.1%
65-74	27,015,122	120.6%	1,014,083,731	99.1%	662,777,279	102.0%	140,088,565	96.2%	2,602,000	127.2%	100,000	44.3%	14,725,682,019	99.9%	36,488,800	108.5%
75-84	5,970,000	330.3%	56,608,495	89.6%	4,065,000	89.8%	0	0.0%					15,797,689,100	100.0%	19,679,355	130.4%
85-99			0	0.0%	10,000	7922.6%							1,439,501,252	99.7%	801,304	109.2%

ia_band1 x iy_band1

ia_band1	iy_band1: 1900-1989		iy_band1: 1990-1999		iy_band1: 2000-2009		iy_band1: 2010+
ia_band1	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
18-24	21,957,490	108.3%	248,374,326	101.8%	249,952,870	116.4%	147,646,395	86.5%
25-34	156,424,882	98.9%	2,415,663,468	99.5%	2,774,706,986	99.2%	920,732,479	103.3%
35-44	323,624,196	104.4%	5,875,528,027	99.7%	8,554,094,758	100.0%	2,237,206,156	101.3%
45-54	340,733,447	101.3%	6,605,000,254	100.2%	10,642,985,328	99.9%	3,040,791,286	99.4%
55-64	443,897,049	104.0%	6,912,220,124	100.0%	8,972,404,806	99.8%	3,118,630,111	99.4%
65-74	172,472,214	106.2%	6,127,207,226	100.2%	8,704,756,697	100.0%	1,604,401,379	98.4%
75-84	5,070,236	115.2%	1,325,459,599	99.5%	13,867,379,988	99.9%	686,102,127	102.4%
85-99	0	0.0%	34,694,322	93.4%	1,279,011,875	99.5%	126,606,359	102.8%

gender x insurance_plan

insurance_plan	gender: F		gender: M
insurance_plan	Outcome	Ratio	Outcome	Ratio
Other	76,199,638	107.7%	111,152,662	96.1%
Perm	4,259,680,761	99.5%	7,902,242,500	100.3%
Term	8,746,195,310	99.2%	30,277,420,743	100.1%
xL	20,175,116,226	100.2%	26,387,728,620	100.0%

gender x ltp

ltp	gender: F		gender: M
ltp	Outcome	Ratio	Outcome	Ratio
5 yr	233,647,378	113.8%	452,696,136	105.3%
10 yr	1,294,444,561	101.9%	5,977,220,162	99.7%
15 yr	1,152,920,343	97.1%	4,864,836,217	100.5%
20 yr	4,067,960,544	98.9%	13,699,088,063	100.2%
25 yr	164,969,732	97.3%	360,253,190	98.8%
30 yr	911,017,939	96.5%	2,181,713,846	100.9%
Not Level Term	24,668,222,437	100.1%	34,946,093,098	100.0%
Unknown	764,009,001	99.3%	2,196,643,813	99.5%

gender x iy_band1

iy_band1	gender: F		gender: M
iy_band1	Outcome	Ratio	Outcome	Ratio
1900-1989	415,833,187	105.2%	1,048,346,327	102.4%
1990-1999	10,124,056,124	100.2%	19,420,091,222	99.9%
2000-2009	19,505,796,137	99.8%	35,539,497,171	100.0%
2010+	3,211,506,487	98.4%	8,670,609,805	100.5%

insurance_plan x ltp

ltp	insurance_plan: Term		insurance_plan: Other		insurance_plan: Perm		insurance_plan: xL
ltp	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
5 yr	686,343,514	108.0%
10 yr	7,271,664,723	100.1%
15 yr	6,017,756,560	99.8%
20 yr	17,767,048,607	99.9%
25 yr	525,222,922	98.4%
30 yr	3,092,731,785	99.5%
Not Level Term	702,195,128	97.4%	187,352,300	100.5%	12,161,923,261	100.0%	46,562,844,846	100.1%
Unknown	2,960,652,814	99.4%

insurance_plan x iy_band1

insurance_plan	iy_band1: 1900-1989		iy_band1: 1990-1999		iy_band1: 2000-2009		iy_band1: 2010+
insurance_plan	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
Other	699,086	15.1%	24,035,150	211.0%	95,671,569	96.6%	66,946,495	93.8%
Perm	685,297,227	99.2%	8,414,471,562	99.9%	2,230,633,550	100.5%	831,520,922	100.9%
Term	110,349,436	113.7%	7,507,786,831	100.1%	24,023,250,728	99.9%	7,382,229,058	99.5%
xL	667,833,765	106.6%	13,597,853,803	99.9%	28,695,737,461	99.9%	3,601,419,817	100.8%

ltp x iy_band1

ltp	iy_band1: 1900-1989		iy_band1: 1990-1999		iy_band1: 2000-2009		iy_band1: 2010+
ltp	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio	Outcome	Ratio
5 yr	11,828,904	103.9%	356,634,986	109.6%	256,413,452	107.5%	61,466,172	102.3%
10 yr	7,313,258	102.3%	395,337,236	102.3%	4,556,557,084	100.1%	2,312,457,145	99.5%
15 yr	770,000	43.9%	786,184,143	100.2%	4,222,555,938	99.6%	1,008,246,479	100.5%
20 yr	27,216,551	142.4%	3,469,245,630	99.6%	11,456,767,739	100.0%	2,813,818,687	99.6%
25 yr	0	0.0%	162,532,838	94.9%	278,353,329	97.4%	84,336,755	109.5%
30 yr	0	0.0%	553,191,034	98.0%	1,883,712,240	100.4%	655,828,511	98.5%
Not Level Term	1,355,860,609	102.4%	22,279,426,292	99.9%	31,476,742,520	99.9%	4,502,286,114	100.7%
Unknown	61,190,192	110.9%	1,541,595,187	100.8%	914,191,006	98.4%	443,676,429	95.7%

Comparison of Model Predictions

Goodness of Fit

It is important to compare model performance on the test dataset. Models tend to fit well on the training data.

We compute the MSE, MAD, and Poisson deviance for each model on the test dataset. Models with lower values are considered qualitatively better.

Across all measures, the elastic net GLM model has the lowest deviation, with the LightGBM qualitatively not far behind. The main-effects GLM does not compete, which reinforces the need for some accommodation of interactions.

#-----------------------------------------#
##### Table of Results #####
#-----------------------------------------#

## creates a table of results for mse, mae, dev for the models

rbind(test[,val(get(resp_var),predictions_glm,get(resp_offset))],
                 test[,val(get(resp_var),predictions_glmnet,get(resp_offset))],
                 test[,val(get(resp_var),predictions_lgbm1,get(resp_offset))]) %>%
  set_rownames(c("glm","glmnet","lgbm")) %>%
  rownames_to_column(var="model") %>%
  as.data.table() %>%
  as_flextable()

model	mse	mae	dev
glm	657,407,396,611,204	10,369,070	1,552,142,707,959,196
glmnet	285,309,606,550,480	8,470,467	723,646,527,740,728
lgbm	355,563,440,625,343	8,741,101	909,129,632,590,165

Graphical Model Comparison

Unlike the GLM, neither the LightGBM nor the penalized GLM provide any information regarding parameter uncertainty. For elastic net GLMs, there are options to estimate parameter uncertainty:

Move to a fully Bayesian setting. This gives the modeler significant control, at the cost of complexity (e.g., how to choose reasonable priors) and computation cost. Stan and INLA are available for this purpose.
Apply the method in Tibshirani et al’s A significance test for the lasso. This requires rerunning penalized GLMs and is thus potentially costly.
Apply the method in Lederer’s Fundamentals of High-Dimensional Statistics, Sec. 5.2. While technically involved, there does not seem to be a heavy computational lift.

To get around the limitations of assessing uncertainty for now, we plot the models versus the envelope of uncertainty arising from the data itself. This shifts the point of view from assessing parameter uncertainty to assessing goodness-of-fit.

Below are plots of how the model performs versus marginal effects, with performance tested on the test subset. Black dots with error bars are from the actual-to-2015VBT ratio, with error bar width based on the dispersion from the GLM model. (Caution: this is at best a crude approximation.)

The following colors denote specific predictive model ratios versus the 2015 VBT:

Red uses GLM predicted claims
Blue uses predicted claims from the elastic net GLM
Green uses LightGBM predicted claims

Broad observations:

In 2017, some of the average relationships shifted versus 2013-2016. This can be seen by noting the model dots resting outside the error bars.
The elastic net model may be missing some higher order interactions.

#-----------------------------------------#
##### Model Feature Plots #####
#-----------------------------------------#

## Select the top `nUseTopLightGBMInteractions` interactions from `imp.int2`
## and transform the `Feature1` and `Feature2` columns into a long format,
## ensuring each feature appears only once in the resulting list.

imp.int2 %>% 
  head(nUseTopLightGBMInteractions) %>%           # Select the top interactions
  select(Feature1, Feature2) %>%                  # Select the relevant columns
  pivot_longer(cols = c(Feature1, Feature2),      # Pivot to long format
               names_to = NULL,
               values_to = "Feature") %>%        
  distinct() %>%                                  # Remove duplicate features
  pull(Feature) -> int.subset                     # Extract the features into `int.subset`

## Calculate the dispersion of the GLM model
glm_disp <- sum(modelGLM$residuals^2 * modelGLM$weights)/modelGLM$df.residual

## Select the top `nUseTopLightGBMInteractions` interactions from `imp.int2` again
## to use as input for the `map2` function
int.subset <- imp.int2 %>% 
  head(nUseTopLightGBMInteractions) %>%
  select(Feature1,Feature2)

## Apply a function to each pair of features in `int.subset` to generate plots
map2(
  .x = int.subset$Feature1,                       # First feature in the pair
  .y = int.subset$Feature2,                       # Second feature in the pair
  .f = function(s1, s2) {                         # Function to apply
    test[,                                        # Subset the data
         .(
           predicted_glm   = sum(predictions_glm)/sum(amount_2015vbt),
           predicted_glmnet= sum(predictions_glmnet)/sum(amount_2015vbt),
           predicted_lgbm1 = sum(predictions_lgbm1)/sum(amount_2015vbt),
           a_e             = sum(amount_actual)/sum(amount_2015vbt),
           stde            = sqrt(sum(amount_actual)*glm_disp)/sum(amount_2015vbt)
         ),
         by=c(s1,s2)]  %>%
      setnames(s1,"x") %>%
      setnames(s2,"byvar") %>%
      mutate(
        x = fct_relevel(
          x,
          sort(levels(x))
        ),
        byvar = fct_relevel(
          byvar,
          sort(levels(byvar))
        )
      ) %>%
      as.data.table() %>%
     ggplot(aes(x = x, y = a_e)) +                # Create a ggplot with `x` on the x-axis and `a_e` on the y-axis
      facet_wrap(vars(byvar)) +                   # Create separate panels for each value of `byvar`
      geom_point() +                              # Add points for actual values
      geom_errorbar(aes(ymin = a_e - 1.96 * stde, ymax = a_e + 1.96 * stde)) + # Add error bars
      geom_hline(yintercept = 1, linetype = 2) +  # Add a horizontal line at y = 1
      geom_point(aes(y = predicted_glm), color = "red") +   # Add points for GLM predictions
      geom_point(aes(y = predicted_glmnet), color = "blue") + # Add points for GLMNet predictions
      geom_point(aes(y = predicted_lgbm1), color = "green") + # Add points for LightGBM predictions
      scale_y_continuous(name = "Factor", labels = scales::percent, trans = "log") + # Log scale for y-axis
      scale_x_discrete(name = s1) +               # Name the x-axis after the first feature
      theme_minimal() +                           # Use a minimal theme
      theme(axis.text.x = element_text(angle = 90, vjust = 0.5, hjust=1))  # Rotate x-axis labels
  }
) %>%
  purrr::set_names(
    int.subset %>% 
      mutate(Features=paste0( Feature1, " x ", Feature2)) %>% 
      select(Features) %>%
      unlist
    ) %>%
  iwalk(~ {
    cat('### ', .y, '\n\n')   # Print the feature pair as a section header
    print(.x)                 # Print the plot
    cat('\n\n')               # Add some spacing
  } )

ia_band1 x uw

face_amount_band x uw

face_amount_band x ia_band1

dur_band1 x face_amount_band

dur_band1 x uw

dur_band1 x ia_band1

face_amount_band x gender

dur_band1 x ltp

face_amount_band x ltp

ltp x uw

face_amount_band x iy_band1

ia_band1 x ltp

insurance_plan x uw

gender x ia_band1

ia_band1 x iy_band1

gender x uw

face_amount_band x insurance_plan

ia_band1 x insurance_plan

iy_band1 x uw

dur_band1 x gender

insurance_plan x ltp

dur_band1 x iy_band1

dur_band1 x insurance_plan

gender x insurance_plan

iy_band1 x ltp

insurance_plan x iy_band1

gender x iy_band1

gender x ltp

Mortality Differences by Product Use Case

Observations from the Raw Data

When assessing differences by product, it is not hard to find challenges when looking at the raw, unadjusted data.

One example is that there is virtually no 4-class non-smoker exposure in Perm, while there is significant exposure in Term. This implies that there is a potential issue of identifiability in interactions between insurance plan and underwriting due to the imbalance in exposures. This manifests as an apparent instability in calibrations.

For example, the marginal difference between the 4-class and 2-class non-smokers in the data is 82.42% (78.2%/94.9%), while the marginal difference between Term and Perm is 86.5% (83.4%/96.4%).

ds[,
             c("Smoker_Status","NClasses","Class"):=tstrsplit(uw,"/")]

ds[Smoker_Status=="N",.(A_2015VBT=sum(get(resp_var))/sum(get(resp_offset))),
   by=.(NClasses)][order(NClasses)] %>%
  flextable() %>%
  set_header_labels(NClasses="No. of Pref. Classes",
                    A_2015VBT="A/2015VBT") %>%
  set_formatter(values = function(x) {
        if(is.numeric(x))
          sprintf( "%.1f%%", x*100 )
        else
          x
        }
        )

No. of Pref. Classes	A/2015VBT
1	97.2%
2	94.9%
3	81.4%
4	78.2%

ds[,.(A_2015VBT=sum(get(resp_var))/sum(get(resp_offset))),by=.(insurance_plan)] %>%
  flextable() %>%
  set_header_labels(insurance_plan="Insurance Plan",
                    A_2015VBT="A/2015VBT") %>%
  set_formatter(values = function(x) {
        if(is.numeric(x))
          sprintf( "%.1f%%", x*100 )
        else
          x
        }
        )

Insurance Plan	A/2015VBT
Term	83.4%
xL	89.2%
Perm	96.4%
Other	84.1%

By way of comparison, the GLM calibrates 86.8% for Term versus Perm, and the weighted average factors for 4-class systems from the GLM model is 79.3% versus 95.7% for 2-class, for a ratio of 82.8%. The main effects GLM is therefore asserting that both conditions are associated with lower mortality.

For the elastic net GLM, the situation is complicated. All in, there are 52 factors which mention insurance plan, and assessing when perm and term differ is challenging on a bare reading of the factor table.

reformatCoefs(cvfit, pred_cols)  %>%
  filter(Coef != 0) %>%
  select(Feature1Name,
         Feature1Level,
         Feature2Name,
         Feature2Level,
         Coef) %>%
  mutate(Coef=exp(Coef)) %>%
  filter(
    Feature1Name == "insurance_plan" | Feature2Name == "insurance_plan"
  ) %>%
  select(Feature2Name, Feature2Level,
         Feature1Name, Feature1Level,
         Coef) %>%
  arrange(Feature2Name, Feature2Level,
         Feature1Name, Feature1Level) %>%
  flextable() %>%
  set_header_labels(Coef="Factor") %>%
  set_formatter(values = function(x) {
        if(is.numeric(x))
          sprintf( "%.1f%%", x*100 )
        else
          x
        }
        )

For the LightGBM model, there is arguably no interesting mean difference for between Perm and Term.

data.table(
    insurance_plan=train[shp_int_subset,insurance_plan],
    shap=shp$S[,which(names(shp$X)=="insurance_plan")] + shp$baseline,
    response=train[shp_int_subset,get(resp_var)],
    offset=train[shp_int_subset,get(resp_offset)]
) %>%
  group_by(insurance_plan) %>%
  summarize(M_T=sum(exp(shap)*offset)/sum(offset)) %>%
  flextable() %>%
  set_header_labels(insurance_plan="Insurance Plan",
                    M_T="Model / 2015VBT") %>%
  set_formatter(values = function(x) {
        if(is.numeric(x))
          sprintf( "%.1f%%", x*100 )
        else
          x
        }
        )

Insurance Plan	Model / 2015VBT
Other	67.0%
Perm	91.3%
Term	93.3%
xL	102.9%

For class system, the LightGBM model is illustrating a substantial reduction in mean mortality for the 4-class systems relative to 2-class systems.

shaps.uw <- data.table(
    uw=train[shp_int_subset,uw],
    shap=shp$S[,"uw"] + shp$baseline,
    response=train[shp_int_subset,get(resp_var)],
    offset=train[shp_int_subset,get(resp_offset)]
) 

shaps.uw[,
             c("Smoker_Status","NClasses","Class"):=tstrsplit(uw,"/")]

shaps.uw %>%
  group_by(NClasses) %>%
  summarize(M_T=sum(exp(shap)*offset)/sum(offset)) %>%
  flextable() %>%
  set_header_labels(NClasses="No. of Pref. Classes",
                    M_T="Model / 2015VBT") %>%
  set_formatter(values = function(x) {
        if(is.numeric(x))
          sprintf( "%.1f%%", x*100 )
        else
          x
        }
        )

No. of Pref. Classes	Model / 2015VBT
1	104.4%
2	100.9%
3	101.0%
4	86.2%

All of this strongly suggests the need for more sophisticated analysis.

Observations from the GLM

One question of interest to actuaries is why different products have different mortality outcomes. Many things could contribute to the difference, such as UW practice, anti-selection risk level, market segment, etc., and generally, it is hard to quantify their impact. With the GLM model and relevant analysis, we have a possible solution.

Let us revisit the table output with insurance plan as the predictor of interest.

## function that calls individual predictor, e.g., insurance plans. 
mainF(df     = ds,
      model  = modelGLM,
      rf     = "insurance_plan",
      resp   = resp_var,
      offset = resp_offset) %>%
  flextable() %>%
  set_header_labels("rowname" = "") %>%
  set_formatter(values = ~ if(is.numeric(.)) sprintf("%.1f%%", . * 100) else .) %>%
  set_caption(caption = paste0("Weighted Average GLM Factors for Variable: insurance_plan"))

Weighted Average GLM Factors for Variable: insurance_plan
	Other	Perm	Term	xL
amount_2015vbt	0.8411	0.9638	0.8340	0.8923
Factor: insurance_plan	1.0000	0.8749	0.7596	0.9590
Ave Fac: dur_band1	0.9306	0.9098	0.9102	0.9024
Ave Fac: face_amount_band	0.7300	0.7913	0.7339	0.7366
Ave Fac: gender	1.0067	1.0070	1.0082	1.0060
Ave Fac: ia_band1	0.9124	0.9220	0.9325	0.8870
Ave Fac: iy_band1	0.8452	0.9105	0.8636	0.8741
Ave Fac: ltp	0.6733	0.6733	0.8455	0.6733
Ave Fac: uw	0.9790	1.0127	0.9051	1.0025

For illustration, let us select Perm and Term for pairwise comparison. By A/15VBT, Perm (96.4%) seems to have better mortality than Term (83.4%). Is this due to “product differences”?

mainF(df = ds, 
      model  = modelGLM, 
      rf     = "insurance_plan", 
      resp   = resp_var, 
      offset = resp_offset) %>%
  select(rowname, Perm, Term) %>%
  flextable() %>%
  set_header_labels("rowname" = "") %>%
  set_formatter(values = ~ if(is.numeric(.)) sprintf("%.1f%%", . * 100) else .) %>%
  set_caption(caption = paste0("Weighted Average GLM Factors for Variable: insurance_plan in (Perm, Term)"))

Weighted Average GLM Factors for Variable: insurance_plan in (Perm, Term)
	Perm	Term
amount_2015vbt	0.9638	0.8340
Factor: insurance_plan	0.8749	0.7596
Ave Fac: dur_band1	0.9098	0.9102
Ave Fac: face_amount_band	0.7913	0.7339
Ave Fac: gender	1.0070	1.0082
Ave Fac: ia_band1	0.9220	0.9325
Ave Fac: iy_band1	0.9105	0.8636
Ave Fac: ltp	0.6733	0.8455
Ave Fac: uw	1.0127	0.9051

Thanks to the GLM model, we can work with a multiplicative formula for prediction. And, with this elegant structure, we can parse out the impact of each individual predictor and make comparisons. The relative impact is represented by the rate of change.

Of these, the movement in uw is the most influential, with ratio 111.9% for Perm over Term. This means that, if all the other predictors are controlled, the average uw factor on a risk-adjusted basis will make Perm mortality prediction approximately 11.9% higher than that of Term. Other influential drivers from this analysis include face amount band, issue year band, and level term period. This may suggest, if actuaries/modelers want to build a simpler model yet still capture essential impact to mortality outcome, they may consider including at least those predictors in the GLM model.

One should nonetheless look to residuals and distributions to ensure that valuable interactions are not being lost. Part of what we are seeing has to do with the different distributions between Perm and Term of uw and face_amount_band. Perm tends to favor 1- and 2-class risk class systems, while Term tends to favor 3- and 4-class systems. Perm also tends to favor lower face amounts, while Term favors higher face amounts.

## create histogram of 2015 VBT Tabular Claims by Count and UW
ds %>%
  filter(insurance_plan %in% c("Perm","Term")) %>%
  group_by(uw,
           insurance_plan) %>%
  summarize(
            AM_policy=sum(policy_actual)/sum(predictions_glm),
            policy_2015vbt=sum(policy_2015vbt)) %>%
  as.data.table() %>%
  ggplot(aes(x = uw)) +
    geom_bar(aes(y = policy_2015vbt), stat = "identity") +
    facet_wrap(facets = vars(insurance_plan)) +
    scale_y_continuous(labels = scales::number, name = "2015 VBT Tabular Claims by Count") +
    theme_minimal() +
    theme(axis.text.x = element_text(angle = 45))

## create histogram of 2015 VBT Tabular Claims by Count and Face Amount
ds %>%
  filter(insurance_plan %in% c("Perm","Term")) %>%
  group_by(face_amount_band,
           insurance_plan) %>%
  summarize(
            AM_policy=sum(policy_actual)/sum(predictions_glm),
            policy_2015vbt=sum(policy_2015vbt)) %>%
  as.data.table() %>%
  ggplot(aes(x = face_amount_band)) +
    geom_bar(aes(y = policy_2015vbt), stat = "identity") +
    facet_wrap(facets = vars(insurance_plan)) +
    scale_y_continuous(labels = scales::number, name = "2015 VBT Tabular Claims by Count") +
    theme_minimal() +
    theme(axis.text.x = element_text(angle = 90, vjust = 0.5, hjust=1))

In light of what we see for distribution, it is unsurprising that the model fits poorly for the Term subset for smaller face amounts and 1- and 2-class systems, while the model fits the Perm subset poorly for 3- and 4-class systems. Since Perm tends to dominate the lower face amounts, model fit is not nearly as poor there as for the Term subset.

## Create graph of Residuals of Actuals versus Count-based Model by UW

ds %>%
  filter(insurance_plan %in% c("Perm","Term")) %>%
  group_by(uw,
           insurance_plan) %>%
  summarize(
            AM_policy=sum(amount_actual)/sum(predictions_glm),
            policy_actual=sum(amount_actual),
            predictions_glm=sum(predictions_glm),
            amount_2015vbt=sum(amount_2015vbt)) %>%
  as.data.table() %>%
  ggplot(aes(x = uw)) +
    geom_point(aes(y = policy_actual / amount_2015vbt), color = "red") +
    geom_point(aes(y = AM_policy, group = 1)) +
    geom_errorbar(aes(ymin = AM_policy - 1.96 * sqrt(glm_disp / predictions_glm), ymax = AM_policy + 1.96 * sqrt(glm_disp / predictions_glm))) +
    geom_hline(yintercept = 1, color = "blue", linetype = 2) +
    facet_wrap(facets = vars(insurance_plan)) +
    scale_y_continuous(labels = scales::percent, name = "Actual-to-Model Ratio") +
    ggtitle("Residuals of Actuals versus Count-based Model") +
    theme_minimal() +
    theme(axis.text.x = element_text(angle = 90, vjust = 0.5, hjust=1))

## Create graph of Residuals of Actuals versus Count-based Model by Face Amount

ds %>%
  filter(insurance_plan %in% c("Perm","Term")) %>%
  group_by(face_amount_band,
           insurance_plan) %>%
  summarize(
            AM_policy=sum(amount_actual)/sum(predictions_glm),
            policy_actual=sum(amount_actual),
            predictions_glm=sum(predictions_glm),
            amount_2015vbt=sum(amount_2015vbt)) %>%
  as.data.table() %>%
  ggplot(aes(x = face_amount_band)) +
    geom_point(aes(y = policy_actual / amount_2015vbt), color = "red") +
    geom_point(aes(y = AM_policy, group = 1)) +
    geom_errorbar(aes(ymin = AM_policy - 1.96 * sqrt(glm_disp / predictions_glm), ymax = AM_policy + 1.96 * sqrt(glm_disp / predictions_glm))) +
    geom_hline(yintercept = 1, color = "blue", linetype = 2) +
    facet_wrap(facets = vars(insurance_plan)) +
    scale_y_continuous(labels = scales::percent, name = "Actual-to-Model Ratio") +
    ggtitle("Residuals of Actuals versus Count-based Model") +
    theme_minimal() +
    theme(axis.text.x = element_text(angle = 90, vjust = 0.5, hjust=1))

Fitting a main-effects GLM on a dataset is a frequently used first step when modeling any dataset. Analyzing residuals from this model and assessing parameter variability by subgroup can reveal useful patterns for further analysis. It is often the case that interactions of effects are present. While a useful starting point, main-effects models cannot capture such interactions effectively. It is therefore necessary to turn to richer models and approaches.

Observations from the Gradient Boosted Decision Tree

Below are box plots of SHAP values for insurance_plan by the other variables.

imp.int2["insurance_plan" == Feature1 | "insurance_plan" == Feature2] %>% 
    head(nPlotTopInteractions) %>%
    select(Feature1,Feature2) %>%
    pivot_longer(cols=c(Feature1,Feature2),
                 names_to=NULL,
                 values_to="Feature") %>%
    distinct() %>%
    filter(Feature != "insurance_plan") -> 
    int.vars

plist <- ilec_shap_plot(
    shp,
    "insurance_plan",
    setdiff(pred_cols,"insurance_plan"),
    resp_var = resp_var,
    resp_offset = resp_offset,
    train.data = train[shp_int_subset]
  )


plist %>%
  iwalk(~ {
    cat('### ',.y,'\n\n')
    print(.x)
    cat('\n\n')
  } )

main effect: insurance_plan

insurance_plan x uw

insurance_plan x face_amount_band

insurance_plan x dur_band1

insurance_plan x ia_band1

insurance_plan x gender

insurance_plan x ltp

insurance_plan x iy_band1

We have noted the interactions with insurance plan from the LightGBM SHAP values as follows:

Main Effect: Perm and Term shap distributions are qualitatively similar, with the xL class higher.
Interaction with underwriting:
1. Substantial interaction with the “Other” category
2. For Term, some evidence of higher mortality for N/4/3 and N/4/4
Interaction with face amount band:
1. No obvious interactions with Perm and Term
2. Weak evidence for interaction with xL, based on U-shaped pattern in boxplots
Interaction with duration
1. Weak evidence for elevated mortality in early durations for Perm
2. Weak evidence for opposite in early durations for Term
3. Face amounts 1 million and higher for “Other” are plainly different from lower face amount “Other”
Interaction with issue age
1. Evidence for different issue age slope (relative to 2015VBT) for xL based on downward trend in boxplots
2. Weak evidence for slight upward issue age slope (relative to 2015VBT) for Term based on trend in boxplots
Interaction with gender: no obvious interaction
Interaction with level term period:
1. Obviously, no interactions outside of Term
2. Within Term, “Not Level Term” has lower mortality than the other level term types
Interaction with issue year band:
1. Since 1990, there is evidence of an upward trend in mortality for all categories outside of “Other”.

Contrasts of Interactions with Insurance Plan from the Elastic Net Model

The elastic net model encodes interesting interactions of insurance plan with other predictor variables. Graphing the contrast between insurance plan types can reveal patterns which are difficult to see when looking at the bare coefficients or tables of factors.

To do so, we can gather the table of factors which include insurance plan and compute the ratio of the factors versus the factors for term. For example, if the marginal factor for Perm males is 99%, and the factor for Term males is 90%, then the ratio is 110%. These contrasts can then be plotted for both males and females across the plan comparisons, as can be seen in the following graphs.

## Combine the list of glmnet interactions into a data.table and filter relevant interactions
data.table(do.call(rbind, glmnet.int.list)) %>%
  filter((Feature1Name == "insurance_plan" | Feature2Name == "insurance_plan") &
         (Feature1Name != "ltp" & Feature2Name != "ltp")) ->
  ints.with.plan

## Extract interaction feature names related to "insurance_plan"
c(ints.with.plan[Feature1Name == "insurance_plan", Feature2Name],
  ints.with.plan[Feature2Name == "insurance_plan", Feature1Name]) ->
  ints.with.plan

## Map over each interaction feature
ints.with.plan %>%
  map(
    .f = \(x) {
      symx <- sym(x)  # Convert feature name to symbol

      ## Create a table of coefficients for the interaction with "insurance_plan"
      tableCVNetCoefs(train.grid,
                      c(x, "insurance_plan"),
                      "Factor",
                      pred_cols,
                      levellist = list("dur_band1" = "04-05")) %>%
        data.table() ->
        tblFacts

      ## Normalize factors by "Term" and pivot longer for ggplot
      cbind(
        tblFacts[, 1],
        tblFacts[, lapply(.SD, "/", Term), .SDcols = colnames(tblFacts)[-1]]
      ) %>%
        pivot_longer(
          cols = colnames(tblFacts)[-1],
          values_to = "Factor",
          names_to = "comparison"
        ) %>%
        filter(comparison != 'Term') %>%
        mutate(comparison = paste0(comparison, " vs. Term")) %>%
        data.table() ->
        dftmp

      ## Special handling if the feature is "uw"
      if (x == "uw") {
        dftmp %>%
          separate(
            col = uw,
            into = c("Smoker_Status", "NClasses", "Class"),
            sep = "/",
            remove = FALSE
          ) ->
          dftmp

        ## Filter out single class and plot
        dftmp %>%
          filter(NClasses != 1) %>%
          ggplot(aes(x = Class, y = Factor)) +
          geom_line(aes(group = comparison, color = comparison)) +
          scale_y_continuous(labels = scales::percent) +
          facet_wrap(Smoker_Status ~ NClasses, labeller = label_both) +
          theme_minimal() +
          ggtitle(
            "Ratio of Insurance Plan Factor to Term Insurance Plan Factor",
            subtitle = paste0("by ", ints.with.plan[3])
          ) -> p
      } else {
        ## General plot for other features
        dftmp %>%
          ggplot(aes(x = !!symx, y = Factor)) +
          geom_line(aes(group = comparison, color = comparison)) +
          scale_y_continuous(labels = scales::percent) +
          theme_minimal() +
          ggtitle(
            "Ratio of Insurance Plan Factor to Term Insurance Plan Factor",
            subtitle = paste0("by ", x)
          ) +
          theme(axis.text.x = element_text(angle = ifelse(x == "face_amount_band", 45, 0))) +
          scale_color_viridis_d() -> p
      }

      p  # Return the plot
    }
  ) %>%
  purrr::set_names(ints.with.plan) %>%
  iwalk(~ {
    cat('### ', .y, '\n\n')
    print(.x)
    cat('\n\n')
  })

iy_band1

uw

face_amount_band

ia_band1

dur_band1

gender

It appears that the gap between UL/VL/ULSG/VLSG and term has been narrowing with increasing issue year.
xL and Other tend to have a wider spread of factors for face amount than term, while perm has a narrower spread of face amount factors than term.
Perm and xL tend to have flatter slope than Term by issue age, except above issue age 65. Above issue age 65, the slope of Perm and xL diverge.
Perm tends to have higher duration 2 experience than others.
The gender differential for males is narrower for Perm than for term.

A different view helps illustrate the interactions of underwriting and insurance plan. It is easier to see in this view that

The residual standard class of a 2-class non-smoker system for Term is much higher than the others.
The spread for Term and xL in the 4-class non-smoker system is wider than for Perm and Other.

## Create graph of interaction of Underwriting and Insurance Plan

tableCVNetCoefs(train.grid,
                c("uw", "insurance_plan"),
                "Factor",
                pred_cols,
                levellist = list("dur_band1" = "04-05")) %>%
  pivot_longer(cols = c("Perm", "Term", "Other", "xL"), values_to = "Factor", names_to = "insurance_plan") %>%
  separate(col = uw, into = c("Smoker_Status", "NClasses", "Class"), sep = "/", remove = FALSE) %>%
  data.table() -> tblFacts


tblFacts %>%
  ggplot(aes(x = insurance_plan, y = Factor)) +
    geom_line(aes(group = Class, color = Class)) +
    facet_wrap(Smoker_Status ~ NClasses, labeller = label_both) +
    theme_minimal() +
    scale_y_continuous(labels = scales::percent) +
    scale_color_viridis_d() +
    ggtitle(
      "Interaction of Underwriting and Insurance Plan",
      subtitle = "from the elastic net model"
    )

Summary

In this analysis, we explored the application of a predictive modeling framework within actuarial experience studies, focusing on mortality differences by product type in the ILEC dataset. Our analysis revealed several key insights and mortality differentials that can be useful to understand drivers of mortality. This framework and the findings demonstrated the use and considerations required of predictive modeling and underscore its value in making informed actuarial decisions.

Acknowledgements

Working Group Members

This framework is the result of the tireless efforts of members of the Individual Life Experience Committee, including

Philip Adams, FSA (chair)
Cynthia Edwalds, FSA
Brian Holland, FSA
Ed Hui, FSA
Michael Niemerg, FSA
Haofeng Yu, FSA

Society of Actuaries

The authors would like to thank the staff of the Society of Actuaries for their help throughout this project. Many thanks to Korrel Crawford and Pete Miller.

Appendices

Computational Requirements

## view session info for documentation purposes
sessionInfo()

## R version 4.2.3 (2023-03-15)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 22.04.4 LTS
## 
## Matrix products: default
## BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
## LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.20.so
## 
## locale:
##  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
##  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
##  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
##  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
##  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       
## 
## attached base packages:
## [1] parallel  stats     graphics  grDevices utils     datasets  methods  
## [8] base     
## 
## other attached packages:
##  [1] flexlsx_0.2.1      openxlsx2_1.8      MatrixModels_0.5-1 patchwork_1.1.2   
##  [5] shapviz_0.7.0      here_1.0.1         arrow_11.0.0.3     ftExtra_0.5.0     
##  [9] flextable_0.9.1    dtplyr_1.3.1       magrittr_2.0.3     lubridate_1.9.2   
## [13] forcats_1.0.0      stringr_1.5.0      purrr_1.0.1        readr_2.1.4       
## [17] tibble_3.2.1       tidyverse_2.0.0    doParallel_1.0.17  iterators_1.0.14  
## [21] foreach_1.5.2      tidyr_1.3.0        ggplot2_3.4.2      EIX_1.2.0         
## [25] dplyr_1.1.1        glmnet_4.1-7       Matrix_1.5-3       lmtest_0.9-40     
## [29] zoo_1.8-12         data.table_1.14.8  lightgbm_3.3.5     R6_2.5.1          
## [33] pre_1.0.6         
## 
## loaded via a namespace (and not attached):
##   [1] uuid_1.1-0              backports_1.4.1         systemfonts_1.0.4      
##   [4] plyr_1.8.8              splines_4.2.3           mycor_0.1.1            
##   [7] urltools_1.7.3          digest_0.6.31           htmltools_0.5.5        
##  [10] earth_5.3.2             fansi_1.0.4             tzdb_0.3.0             
##  [13] officer_0.6.2           askpass_1.1             timechange_0.2.0       
##  [16] gfonts_0.2.0            colorspace_2.1-0        ggrepel_0.9.3          
##  [19] ggiraphExtra_0.3.0      textshaping_0.3.6       xfun_0.38              
##  [22] crayon_1.5.2            jsonlite_1.8.4          libcoin_1.0-9          
##  [25] survival_3.5-3          glue_1.6.2              gtable_0.3.3           
##  [28] ppcor_1.1               sjmisc_2.8.9            shape_1.4.6            
##  [31] scales_1.2.1            fontquiver_0.2.1        mvtnorm_1.1-3          
##  [34] Rcpp_1.0.10             plotrix_3.8-2           viridisLite_0.4.1      
##  [37] xtable_1.8-4            bit_4.0.5               Formula_1.2-5          
##  [40] fontLiberation_0.1.0    htmlwidgets_1.6.2       RColorBrewer_1.1-3     
##  [43] ellipsis_0.3.2          pkgconfig_2.0.3         farver_2.1.1           
##  [46] sass_0.4.5              utf8_1.2.3              crul_1.3               
##  [49] tidyselect_1.2.0        labeling_0.4.2          rlang_1.1.0            
##  [52] reshape2_1.4.4          later_1.3.0             munsell_0.5.0          
##  [55] TeachingDemos_2.12      tools_4.2.3             cachem_1.0.7           
##  [58] xgboost_1.7.5.1         cli_3.6.1               generics_0.1.3         
##  [61] sjlabelled_1.2.0        broom_1.0.4             evaluate_0.20          
##  [64] fastmap_1.1.1           yaml_2.3.7              ragg_1.2.5             
##  [67] knitr_1.42              bit64_4.0.5             zip_2.3.0              
##  [70] nlme_3.1-162            mime_0.12               ggiraph_0.8.7          
##  [73] xml2_1.3.3              compiler_4.2.3          rstudioapi_0.14        
##  [76] curl_5.0.0              bslib_0.4.2             stringi_1.7.12         
##  [79] plotmo_3.6.2            DALEX_2.4.3             highr_0.10             
##  [82] iBreakDown_2.0.1        gdtools_0.3.3           lattice_0.20-45        
##  [85] fontBitstreamVera_0.1.1 vctrs_0.6.2             pillar_1.9.0           
##  [88] lifecycle_1.0.3         triebeard_0.4.1         jquerylib_0.1.4        
##  [91] insight_0.19.1          httpuv_1.6.9            promises_1.2.0.1       
##  [94] codetools_0.2-19        MASS_7.3-58.2           assertthat_0.2.1       
##  [97] openssl_2.0.6           rprojroot_2.0.3         withr_2.5.0            
## [100] httpcode_0.3.0          mgcv_1.8-42             hms_1.1.3              
## [103] grid_4.2.3              rpart_4.1.19            rmarkdown_2.21         
## [106] inum_1.0-5              partykit_1.2-20         shiny_1.7.4

References

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Fokkema, Marjolein. 2020. “Fitting Prediction Rule Ensembles with R Package pre.” Journal of Statistical Software 92 (12): 1–30. https://doi.org/10.18637/jss.v092.i12.

Gohel, David, and Panagiotis Skintzos. 2023. Flextable: Functions for Tabular Reporting. https://CRAN.R-project.org/package=flextable.

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Toward a Predictive Analytics Framework

P. Adams, C. Edwalds, B. Holland, E. Hui, M. Niemerg, H. Yu

July 10, 2024

Executive Summary

Data

Machine Learning in Mortality Studies

Problem Statement

Methodology

Modeling Approaches

Generalized Linear Models (GLM)

Gradient-Boosted Decision Trees (GBDT)

Model Explanation

Ordered Lorenz Plot and Gini

Lift Plot

SHAP

Feature Importance

Goodness-of-Fit

Framework Preparation

Models

GLM

Model Summary

Model, Table of Coefficients, ANOVA

Model Summary

ANOVA

Lift

Lorenz Plot

Model Illustrations and Graphics

Effects Plots

uw

face_amount_band

dur_band1

ia_band1

gender

insurance_plan

ltp

iy_band1

Goodness-of-Fit

Unvariate Goodness-of-Fit

uw

face_amount_band

dur_band1

ia_band1

gender

insurance_plan

ltp

iy_band1

Bivariate Goodness-of-Fit

uw x face_amount_band

uw x dur_band1

uw x ia_band1

uw x gender

uw x insurance_plan

uw x ltp

uw x iy_band1

face_amount_band x dur_band1

face_amount_band x ia_band1

face_amount_band x gender

face_amount_band x insurance_plan

face_amount_band x ltp

face_amount_band x iy_band1

dur_band1 x ia_band1

dur_band1 x gender

dur_band1 x insurance_plan

dur_band1 x ltp

dur_band1 x iy_band1

ia_band1 x gender

ia_band1 x insurance_plan

ia_band1 x ltp

ia_band1 x iy_band1

gender x insurance_plan

gender x ltp

gender x iy_band1

insurance_plan x ltp

insurance_plan x iy_band1

ltp x iy_band1

Subgroup Variability

uw

face_amount_band

dur_band1

ia_band1